ARTICLE TEMPLATE
Rough Shapley unc ions o games wi h a p io i unions
M. Basallo ea, J.M. Galla doa, C. He n´andez-Mance aaand A. Jim´enez-Losadaa
aDepa men o Applied Ma hema ics II, ETSI, Uni e si y o Se ille, Spain
ARTICLE HISTORY
Compiled Janua y 12, 2024
ABSTRACT
A amily o alloca ion ules o coope a i e games wi h a p io i unions is in oduced
in his pape . These alloca ion ules, which will be called ough Shapley alues,
a e ex ensions o he well-known Shapley alue o classical coope a i e games. The
amily o ough Shapley alues, which is cons uc ed by using ough se s o desc ibe
di e en coope a i e op ions, includes se e al o he ex ensions o he Shapley alue
ound in he li e a u e. We p o e ha he ough Shapley alues a e he alloca ion
ules o games wi h a p io i unions ha sa is y some easonable p ope ies.
KEYWORDS
game heo y; coope a i e games; ough se s; Shapley alue; a p io i unions
1. In oduc ion
The heo y o coope a i e games wi h ans e able u ili y s udies si ua ions in which a
g oup o agen s coope a e o ob ain some p o i . One o he main goals o his heo y
is o p o ide ai alloca ion ules o dis ibu ing he p o i among he agen s. The
Shapley alue [17] is he mos commonly used alloca ion ule. Bu he use o he
Shapley alue equi es symme y in he ela ionships be ween he agen s. Howe e ,
he e a e o en special alliances o incompa ibili ies which condi ion he way in which
he agen s in e ac . These si ua ions equi e speci ic models. The model on which his
a icle is based is ha o coope a i e games wi h a p io i unions, in oduced by Owen
[13] in 1977, which is ocused on he s udy o coope a i e si ua ions whe e he e a e
p io g oup alliances be ween agen s. Such alliances a e desc ibed by pa i ions in he
agen se . Owen’s model has been gene alized using o he ypes o s uc u es, such
as coali ion con igu a ions [1], p oximi y ela ions [5] o colo ed g aphs [12]. Owen
ob ained and cha ac e ized a alue o games wi h a p io i unions. This alue ex ends
he Shapley alue in he sense ha in he pa icula case ha he e a e no p e ious
alliances, i is equal o he Shapley alue. Since 1977, mul iple ex ensions o he Shapley
alue ha e been ob ained o games wi h a p io i unions (see, o ins ance, [2,3,11]).
Besides, modi ica ions o he Shapley alue o a p io i unions we e gi en by B ink
and Die z [4] and by Gon¸cal es-Dosan os and Alonso-Meijide [8].
Rega ding Owen’s model o coope a i e games wi h a p io i unions, he easons o
he o ma ion o p io g oups can be a ious, such as in e es s ela ed o he game i sel ,
ex e nal in e es s, amily o iendship ela ionships o geog aphical p oximi y. I is a
CONTACT A. Jim´enez-Losada. Email: a[email p o ec ed]
his poin ha he mo i a ion o he p esen pape a ises. I seems easonable ha
he na u e o he p e ious alliances will in luence how playe s om di e en coali ions
coope a e. Thus, a g oup may ac coe ci ely ( his is he case o he si ua ions s udied in
[9]), p e en ing i s membe s om coope a ing unless he en i e g oup pa icipa es, o
i may be indi e en , o i could e en encou age i s membe s o coope a e wi h o he
agen s, as long as he whole g oup bene i s om such coope a ion. These di e en
beha io s should a ec he alloca ion ule. In o de o o mally desc ibe he di e en
possible beha io s o he p io g oups, we p opose o use ough se s, in oduced by
Pawlak [14] in 1982. In each se o agen s (coali ion) we will conside he pa i ion
induced by he p io g oups, and hen we will use some coe icien s o desc ibe he
beha io o he se s in his pa i ion. In game heo y, ough se s ha e been used mos ly
in non-coope a i e games (see, o ins ance, [10]). In coope a i e games, Polkowski
[15] used hem o es ima e he wo hs o ce ain coali ions in games wi h incomple e
in o ma ion. In his pape we use ough se s o ob ain a amily o alues o coope a i e
games wi h a p io i unions. These alues a e ex ensions o he Shapley alue. The
me hod ha we use is cons uc i e and i allows a pa ame ic desc ip ion o he
alues in he amily, simila ly o how Radzik and D iessen [18] desc ibe a amily o
solu ions o s anda d coope a i e games. Besides, we in oduce a new app oach o
ob ain solu ions o coope a i e games wi h a p io i unions, in he sense ha we do
no use a model based on a wo-s ep nego ia ion.
This pape is o ganized as ollows. In Sec ion 2 we ecall some p elimina ies con-
ce ning coope a i e games and games wi h a p io i unions. In Sec ion 3 we p esen an
axioma ic desc ip ion o a amily o ex ensions o he Shapley alue o games wi h
a p io i unions. In Sec ion 4, ano he amily o ex ensions o he Shapley alue is de-
sc ibed. In his case he desc ip ion is no axioma ic. Ins ead, his amily is ob ained
by applying he Shapley alue o a pa ame ic modi ica ion o he unde lying game
aking in o accoun he a p io i unions. Finally in Sec ion 5 we show he equi alence
be ween he amilies ob ained in Sec ions 3 and 4.
2. Games wi h a p io i unions
Le Nbe a ini e se wi h |N|=n. A coope a i e game on Nis de ined by a cha -
ac e is ic unc ion : 2N→Rwhe e 2Nis he se o all subse s o Nand (∅) = 0.
The elemen s o Na e called playe s, he subse s S⊆Ncoali ions and (S) is he
wo h o S, ha is, he collec i e paymen ha he playe s in Swould ob ain i he
coope a e. We deno e by GN he amily o coope a i e games wi h se o playe s N.
The se o games GNis a eal ec o space wi h dimension 2n−1 i we conside he
ope a ions +wand a wi h , w ∈GNand a∈Rgi en by ( +w)(S) = (S)+w(S)
and (a )(S) = a (S) o e e y S⊆N. I T∈2N {∅} he unanimi y game o Tis he
game uT∈GNde ined by uT(S) = 1 i T⊆Sand uT(S) = 0 o he wise. Unanimi y
games o m a basis o GN, ha is, e e y game ∈GNcan be w i en as
=X
{T⊆N:T6=∅}
∆
TuT,(1)
whe e he numbe s ∆
Ta e called he Ha sanyi coe icien s o . Playe i∈Nis said
o be null in ∈GNi (S) = (S {i}) o e e y S⊆N. I θis a pe mu a ion
o N hen we deno e θi =θ(i) o each i∈Nand θS ={θi :i∈S} o each
S⊆N. I ∈GN hen he pe mu ed game is he game θ ∈GNwi h θ (θS) = (S)
2
o e e y S⊆N. A payo ec o o a game ∈GNis any ec o x∈RN. The
componen xiis in e p e ed as he paymen o playe i∈N. A alue on GNis a
unc ion N:GN→RN, which assigns a payo ec o N( ) o each game ∈GN.
The bes -known alue o coope a i e games is he Shapley alue [17], φN. Fo each
game ∈GN he payo o a playe i∈Nis de ined by
φN
i( ) = X
{S⊆N:i∈S}
γn
s[ (S)− (S {i})] ,wi h γn
s=(s−1)!(n−s)!
n!.(2)
whe e s=|S|. The Shapley alue sa is ies he ollowing p ope ies: (S1) e iciency: i
∈GN, hen Pi∈NφN
i( ) = (N); (S2) linea i y: i , w ∈GNand a, b ∈R, hen
φN(a +bw) = aφN( ) + bφN(w); (S3) null playe p ope y: i i∈Nis a null playe
o ∈GN, hen φN
i( ) = 0; (S4) symme y: i θis a pe mu a ion o Nand ∈GN,
hen φN
θi(θ ) = φN
i( ) o e e y i∈N. No ice ha symme y implies equal ea men
o symme ic playe s, ha is, i i, j ∈N, ∈GNand (S∪{i}) = (S∪{j}) o all
S⊆N {i, j}, hen φi( ) = φj( ). These p ope ies cha ac e ize he Shapley alue,
ha is, φis he unique alue on GN ha sa is ies linea i y, e iciency, null playe
p ope y and symme y. A solu ion unc ion o coope a i e games is a mapping
ha assigns a alue Non GN o each ini e se N. The Shapley unc ion is he
solu ion unc ion φ ha assigns o each ini e se o playe s N he Shapley alue on
N. I ∈GNand N0⊂N, hen he game ∈GN0is he cha ac e is ic unc ion
es ic ed o 2N0. The Shapley unc ion sa is ies his p ope y: (S5) i Nis a ini e
se o playe s and i∈Nis a null playe in ∈GN, hen φN
j( ) = φN {i}
j( ) o all
j∈N {i}.
Owen [13] in oduced games wi h a p io i unions. S a ing om he idea ha in
a coope a i e si ua ion some playe s a e mo e likely o ac oge he han o he s, he
conside ed p io alliances in he se o playe s. An a p io i unions s uc u e o pa i ion
is a amily Po disjoin nonemp y se s o playe s, namely a pa i ion o a ce ain se
N(P) = SA∈P A. In ou case N=N(P). Each elemen A∈ P is called a union o
g oup and i is in e p e ed as a se o playe s wi h common in e es s. A pa i ion P
is indi idualis i |A|= 1 o all A∈ P, and i is globalis i |P| = 1. A coali ional
solu ion unc ion is a mapping ha assigns a alue Pon GN o each pa i ion P. In
pa icula , all solu ion unc ions a e coali ional solu ion unc ions which do no ake
in o accoun he pa i ion, ha is, P( ) = P0( ) i N(P) = N(P0) = N. Owen [13]
desc ibed a wo-s ep me hod o ob ain a coali ional solu ion unc ion, which u ned
ou o be an ex ension o he Shapley unc ion. Le P={A1, ..., Am}be a pa i ion
and ∈GN. Le us deno e M={1, ..., m}. The quo ien game P∈GM o is gi en
by P(Q) = N(PQ) o all Q⊆Mwhe e PQ={Aq:q∈Q}. Fo each S⊆A∈ P
we de ine he es ic ed pa i ion PS= (P {A})∪{S}. A game wq∈GAq o each
q∈Mis now in oduced as wq(S) = φM
q( PS) o all S⊆Aq. Finally, he Owen
unc ion is de ined as he coali ional solu ion unc ion ψwhe e o e e y pa i ion P,
∈GNand i∈N,
ψP
i( ) = φAq
i(wq),(3)
whe e Aqis he g oup in Psuch ha i∈Aq. In bo h globalis and indi idualis
pa i ions he Owen unc ion and he Shapley unc ion coincide, ha is, ψP( ) =
φN( ) o all ∈GN. The Owen alue ψPsa is ies (S1-S3) bu no (S4). The Owen
unc ion has wo main c i icisms: i sa is ies he null playe ( hus denying a easonable
3
payo o any null playe ha is pa o a coali ion wi h a leas one non-null playe )
and he equal ea men o he indi idualis pa i ion and he globalis pa i ion (i
does no di e en ia e be ween hese wo ex eme cases).
3. Coali ional Shapley unc ions
Se e al ex ensions o he Shapley alue o games wi h a p io i unions ha e been
s udied in he li e a u e [2–4,8,11]. No all o hem e ain he essence o he Shapley
alue. Fo example, ex ensions ha e been de ined ha a e no linea o ha p o ide
only g oup payo s (no indi idual ones). Wi h he idea o desc ibing a amily o
coali ional solu ion unc ions, we p opose an axioma ic de ini ion ha ensu es ha we
main ain undamen al p ope ies o he Shapley alue. In addi ion, we ha e sea ched
o axioms ha a e sa is ied by he majo i y o ex ensions o he Shapley alue o
games wi h a p io i unions in oduced in he li e a u e.
We p opose he ollowing axioms o a coali ional solu ion unc ion .
E iciency. I Pis a pa i ion and ∈GN, hen X
i∈N
P
i( ) = (N).
Linea i y. I Pis a pa i ion, a, b ∈Rand , w ∈GN, hen P(a +bw) = a P( ) +
b P(w).
Le Pbe a pa i ion. A pe mu a ion θo e Nis P-compa ible i θA ∈ P o all
A∈ P.
Compa ible symme y. I θis a pe mu a ion P-compa ible o e N o a pa i ion
P, hen P
θi (θ ) = P
i( ) o all ∈GN.
Obse e ha compa ible symme y implies equal ea men o symme ic playe s
i, j ∈Nin i i, j a e in he same g oup, in he sense ha P
i( ) = P
j( ) ( his is
called in acoali ional symme y in [7]). Le Pbe a pa i ion. The se A∈ P is said o
be a null g oup o ∈GNi any i∈Ais a null playe o game . The ollowing axiom
is a sligh modi ica ion o he p ope y o independence o null coali ions in oduced
in [7].
Null g oup p ope y. Le Pbe a pa i ion. I A∈ P is a null g oup o ∈GN
hen
P
i( ) = 0,i i∈A,
P A
i( ),o he wise.
I we ake an indi idualis ic pa i ion hen he axioms abo e co espond o he
classical axioms (S1-S4) o he Shapley alue plus condi ion (S5).
De ini ion 3.1. A coali ional solu ion unc ion is said o be a coali ional Shapley
unc ion i i sa is ies e iciency, linea i y, compa ible symme y and null g oup p op-
e y.
The Shapley unc ion and he Owen unc ion a e examples o coali ional Shapley
unc ions. O he examples a e he alues in oduced in [11] and [3]. The p opo ional
Shapley alue in oduced in [2] is no a coali ional Shapley unc ion since i is no
linea . La e we will show ha he e a e mo e coali ional Shapley unc ions.
In he ollowing sec ion we p esen a cons uc i e me hod ha allows o ob ain a
amily o coali ional Shapley unc ions. Wi h his me hod we will achie e he ollowing:
4
•A pa ame ic desc ip ion o he amily, simila o he desc ip ion ob ained by
Radzik and D iessen [18] o a amily o solu ions o s anda d coope a i e games.
•A new app oach o ob ain solu ions o games wi h a p io i unions, which is no
based on a wo-s ep nego ia ion.
•A way o modeling he a i ude o he g oups in he nego ia ion.
4. Rough Shapley unc ions
In he app oach p oposed by Owen [13] o ob ain a solu ion o games wi h a p io i
unions, wo assump ions s and ou : 1) in o de o de e mine he collec i e payo o a
p io coali ion, all he p io coali ions mus ac en bloc, and 2) in o de o de e mine
he payo o each playe in a p io coali ion, he es o he p io coali ions mus ac
en bloc. The e o e, i he pa i ion o Nis P={A1, ..., Am}, he only coali ions ha
ma e in Owen’s model a e hose o he o m Ai1∪. . . ∪Aik−1∪Ewhe e E⊆Ak.
The es o he coali ions a e o ally igno ed. This, in addi ion o en ailing loss o
in o ma ion, seems somewha a bi a y, since i p esupposes ce ain a i udes o he
playe s owa ds coali ion o ma ion. Ou goal is o in oduce a b oade model ha
allows o conside a ange o a i udes o he playe s, and in which Owen’s model is a
pa icula case. Thus, in his pape , in o de o ob ain a coali ional Shapley alue, we
will ha e o somehow desc ibe how pa i ioning a ec s coali ion gains.
4.1. Rough e sions o a game wi h a p io i unions
We p opose o use ough se heo y [14] in he con ex o coope a i e games wi h a
p io i unions. Le Pbe a pa i ion and N=N(P). I S⊆N hen he in e io and
he closu e o Sin Pa e, espec i ely,
˚
S=[
{A∈P:A⊆S}
Aand ¯
S=[
{A∈P:A∩S6=∅}
A. (4)
Coali ion Sis said o be c isp i ˚
S=¯
S, ha is, i i is he union o g oups in he
pa i ion. Any non-c isp coali ion is called a ough coali ion.
Suppose ha he playe s in a coali ion Sin end o coope a e. I Sis c isp, hen i
is clea all he playe s in Scan coope a e wi hou any es ic ions and he playe s in
N Swill play no ole. I Sis no c isp, hen he si ua ion is di e en , since he playe s
in S ha a e no in ˚
Sdo no ha e comple e eedom o coope a e wi hin S, since hey
a e in p io g oups which a e no con ained in S, and, he e o e, he playe s in hese
g oups could impede hei coope a ion. In his case, ollowing ough se heo y, we will
use he lowe app oxima ion (i.e., he in e io ) and he uppe app oxima ion (i.e., he
closu e) o S, and we will use some coe icien s o exp ess he paymen achie able by
coali ion S( aking in o accoun he a p io i union s uc u e) as a linea combina ion
o he paymen s (igno ing he a p io i union s uc u e) o he coali ions in he in e al
[˚
S, ¯
S] = {R:˚
S⊆R⊆¯
S}. The coe icien s o his linea combina ion will be deno ed
αS
R o each R∈[˚
S, ¯
S]. The se o all coe icien s will be α=nαS
RR∈[˚
S, ¯
S]:S⊆No.
In his way we can conside a new game αas
α(S) = X
R∈[˚
S, ¯
S]
αS
R (R) (5)
5

o each S⊆N. I is clea ha his cons uc ion is oo gene al, and ha we need o
equi e condi ions on he coe icien s αS
R.
De ini ion 4.1. Le Pbe a pa i ion o a ini e se N. A ough coe icien o Pis
a se o eal numbe s α=nαS
RR∈[˚
S, ¯
S]:S⊆No ha , o any A∈ P, sa is ies he
ollowing h ee condi ions:
1) Sha pness: αA
A= 1.
2) In a iance: αS∪A
R∪A=αS
R o all S⊆N Aand R∈[˚
S, ¯
S].
3) Comple eness: i S⊆N A,R∈[˚
S, ¯
S] and S(T(S∪A, hen
X
L∈[∅,A]
αT
R∪L=αS
R.
The amily o ough coe icien s o Pis deno ed by RCP. Fo each α∈RCPand
∈GN, he game αde ined by (5) will be called α- ough e sion o .
Fo any pa i ion, he amily o ough coe icien s is nonemp y, since we can ake
αS
S= 1 o all S⊆Nand he es o numbe s equal o ze o. This ough coe icien is
called he c isp coe icien and i is deno ed by δ. I is clea ha δ= o any coop-
e a i e game ∈GN. Mo eo e i Pis indi idualis ic hen he only ough coe icien
is he c isp coe icien and hence he only ough e sion o he game is i sel .
Ou aim now is o jus i y he condi ions in De ini ion 4.1. The ollowing p oposi ion
shows ha he α- ough e sions o coope a i e games a e he games αwhich do no
modi y ei he he paymen s o c isp coali ions o he cha ac e o null playe s which
a e con ained in null g oups.
P oposi ion 4.2. Le Pbe a pa i ion o a ini e se N. The numbe s α=
nαS
RR∈[˚
S, ¯
S]:S⊆No o m a ough coe icien i and only i , o e e y coope a i e
game ∈GN, he ollowing condi ions hold:
1) α(S) = (S) o e e y c isp coali ion S⊆N,
2) i i∈A∈ P and all playe s in Aa e null playe s in , hen iis null in α.
P oo . Fi s ly we will ake a ough coe icien αand p o e ha he condi ions abo e
a e sa is ied.
1) I Sis c isp hen Sis a union o g oups, ha is, S=A1∪. . . ∪Akwi h
A1, . . . , Ak∈ P. By sha pness, αA1
A1= 1. By in a iance, i ollows ha αA1∪A2
A1∪A2=
1. Successi ely applying he in a iance p ope y we ob ain ha αS
S= 1. Since
[˚
S, ¯
S] = {S}, we ha e α(S) = αS
S (S) = (S).
2) Suppose ha A∈ P and all playe s in Aa e null playe s in . I T={i1, ..., ip} ⊆
A, hen, o all R⊆N,
(R) = (R∪{i1}) = (R∪{i1, i2}) = ··· = (R∪{i1, ..., ip−1}) = (R∪T).(6)
Le S, T ⊆Nbe such ha S⊆N Aand T⊆A. We will p o e ha
α(S∪T) = α(S).(7)
6
We can suppose ha Tis nonemp y. I T=A hen A⊆˚
S∪Aand hence we ha e
[˚
S∪A, ¯
S∪A] = {R∪A:R∈[˚
S, ¯
S]}. Fo each R∈[˚
S, ¯
S] we ha e αS∪A
R∪A=αS
R
by he in a iance p ope y o α, and also (R∪A) = (R) by (6). The e o e,
α(S∪A) = X
R∈[˚
S, ¯
S]
αS∪A
R∪A (R∪A) = X
R∈[˚
S, ¯
S]
αS
R (R) = α(S).
Now we conside T6=A. In his case, we ob ain he in e al
[˚
S∪T, ¯
S∪T] = {R∪L:R∈[˚
S, ¯
S], L ∈[∅, A]}.
By comple eness, each R∈[˚
S, ¯
S] sa is ies PL∈[∅,A]αS∪T
R∪L=αS
R. We also ha e
ha (R∪L) = (R) by (6). The e o e,
α(S∪T) = X
R∈[˚
S, ¯
S]X
L∈[∅,A]
αS∪T
R∪L (R∪L) = X
R∈[˚
S, ¯
S]
(R)X
L∈[∅,A]
αS∪T
R∪L
=X
R∈[˚
S, ¯
S]
(R)αS
R= α(S).
Finally we ake i∈Aand we p o e ha iis a null playe in α. Le S⊆N {i}.
I we apply (7) o he pai (S A, (S∩A)∪{i}) and also o he pai (S A, S ∩A),
we ob ain
α(S∪{i}) = α(S A) = α(S).
Now le α=nαS
RR∈[˚
S, ¯
S]:S⊆Nobe such ha condi ions 1 and 2 desc ibed in
he p oposi ion a e sa is ied. Le us p o e ha αis a ough coe icien .
Take A∈ P.
Since Ais c isp, we ha e, by condi ion 1, ha uα
A(A) = uA(A) = 1. Besides, by
de ini ion, uα
A(A) = αA
AuA(A) = αA
A. We conclude ha αsa is ies sha pness.
Now conside S⊆N Aand ∅ 6=S0⊆A. We aim o check ha i R∈[˚
S, ¯
S], hen
X
L∈[˚
S0,¯
S0]
αS∪S0
R∪L=αS
R.(8)
We will p oceed by induc ion on d(R) = |¯
S|− |R|. We will use he unanimi y game
uR. No ice ha all playe s in Aa e null playe s in uR.
Base case. I d(R) = 0 (i.e., R=¯
S) we ake he game u¯
S. By condi ion 2 all playe s
in S0a e null playe s in uα
¯
S. F om (6) and de ini ion (5) we ob ain
X
L∈[˚
S0,¯
S0]
αS∪S0
¯
S∪L=uα
¯
S(S∪S0) = uα
¯
S(S) = αS
¯
S.
Induc i e s ep. Suppose ha (8) is ue o d(R)< k. Le us check ha hen i is
ue o d(R) = k. I we calcula e uα
R(S∪S0) and uα
R(S) and ake in o accoun (6) and
7
condi ion 2, we ob ain
X
R0∈[R, ¯
S]X
L∈[˚
S0,¯
S0]
αS∪S0
R0∪L=uα
R(S∪S0) = uα
R(S) = X
R0∈[R, ¯
S]
αS
R0.(9)
Fo each R0∈[R, ¯
S] {R}we ha e ha d(R0)< k and we can apply he induc ion
hypo hesis. The e o e, PL∈[˚
S0,¯
S0]αS∪S0
R0∪L=αS
R0. F om his and (9) we conclude ha
X
L∈[˚
S0,¯
S0]
αS∪S0
R∪L=αS
R.
We ha e p o ed (8). Now, i we apply his equali y o S0=A hen [˚
S0,¯
S0] = {A}and
we ob ain in a iance. And i we apply (8) o S0∈[∅, A] {∅, A} hen [˚
S0,¯
S0] = [∅, A]
and, by aking T=S∪S0, we ob ain comple eness.
4.2. The magic cube o a ough coe icien
In his sec ion we will iden i y a ough coe icien wi h a magic cube. Then we will
sol e he sys em o equa ions ha de e mines a ough coe icien . This will gi e us
in o ma ion abou he dimension o he amily o ough coe icien s.
Le N={1, ..., n}and P={A1, ..., Am}. We assume ha we ha e a coali ion
Sas S=S1∪ ··· ∪ Smwhe e Sk(Akis a nonemp y subse in each g oup1. The
comple eness condi ion says
X
R⊆Ak
αS
Sm
p=1,p6=kTp∪R=αS Sk
Sm
p=1,p6=kTp
o any k= 1, ..., m and Tp⊆Apwi h p= 1, ..., m,p6=k. In a simila way, using again
he comple eness p ope y,
X
R⊆Ak
αS Sq∈QSq
Sq∈QTq∪R=αS Sq∈Q∪kSq
Sq∈QTq
o all nonemp y Q⊆ {1, ..., m}, wi h k /∈Qand wi h ca dinali y |Q|< m −1. I
|Q|=m−1, namely Q={1, .., m} {k}, hen we ac ually ha e
X
R⊆Ak
αSk
∅∪R=α∅
∅= 1.
Thus we ob ain a magic cube s uc u e. Figu e 1 shows his idea wi h h ee g oups.
We use he no a ion αS
R1R2R3 o ep esen αS
R1∪R2∪R3wi h Rk⊆Ak o k= 1,2,3.
We aim o calcula e he dimension o he se o ough coe icien s. We will sol e
he linea sys em o med by he condi ions o he de ini ion o ough coe icien o a
gi en coali ion S. In a iance implies again ha we can assume S=S1∪···∪Smwi h
∅ 6=Sk(Ak o e e y k= 1, ..., m. Le |Ak|=akand conside a1≤ ··· ≤ am. Ou
unknowns a e he coe icien s αS
Rwi h R⊆N. Fo his goal, we will desc ibe he sys em
1I Sk=Ak o some k hen he in a iance condi ion implies ha his g oup can be emo ed.
8
α∅∅∅
#$$$$ α∅%&∅
#$$$$$$
α∅%&%'
#$$$$$$$
α%(%&∅
#$$$$$$$
α%(∅∅
#$$$$$$$
α%(%&%'
#$$$$$$$$$
α%(∅%'
#$$$$$$$$$
α∅∅%'
#$$$$$$$
α%(∅
# #&
α∅∅
# #&
α%(%'
# #&
α∅%'
# #&
α
∅∅
#
#
'
α
∅%&
#
#
'
α%(%&
# #'
α%(∅
# #'
α∅∅
# #(α%&∅
# #(
α∅%'
# #(α%&%'
# #(
1
α
∅
#$
α
%
$
#$
α∅
#&
α%&
#&
α∅∅
# #(
α∅%&
# #(
Figu e 1. 3D ough magic cube.
as a ma ix using he ollowing o de s o unknowns and equa ions. We conside each
amily 2Ako de ed by he size and he labels. Fo example, i A1={1,2,3} hen we
ake he o de in 2A1:{∅,1,2,3,12,13,23, A1}. The unknowns a e o de ed as ollows.
Fi s we se a subse o A1, hen one o A2, and so on un il we each Amwhe e we
a e se all subse s o i using he abo e in e nal o de in 2Am. We epea he p ocess
by changing he elemen ixed in Am−1, la e wi h Am−2an so on un il A1. Le ’s look
a he idea in he h ee-g oup example. In his case, we go h ough he aces in he
cube ha ace us ( hose o he di ec ion A1) om he bo om owa ds us; in each ace,
om he le column o he igh ; and, in each column, om op o bo om:
αS
∅∅∅, ..., αS
∅∅A3, ..., αS
∅A2∅, ..., αS
∅A2A3, ..., αS
A1∅∅, ..., αS
A1∅A3, ..., αS
A1A2∅, ..., αS
A1A2A3.
The numbe o unknowns is 2Pm
k=1 ak. We o de he equa ions ans e sely o he
unknowns. Fi s we pu he equa ions unning h ough A1wi h he o he g oups wi h
ixed coali ions (again aking in o accoun he in e nal o de o he g oups). Then he
ones ha un h ough A2and so on. Fo example, in he igu e he i s equa ions a e
9
by he educ ion p ope y o α, ha
α(P)(S) = X
T∈[˚
S, ¯
S]
αS
T(P) (T) = X
T∈[˚
S, ¯
S]
αS
T(P A) (T) = α(P A)(S)
whe e [˚
S, ¯
S] is he same in Pand P Abecause S⊆N A. Le A={i1, ..., ip}. We
p o ed be o e ha i1, ..., ipa e null playe s in α(P). By (S5),
φP,α
i( ) = φN
i( α(P)) = φN {i1}
i( α(P)) = ··· =φN A
i( α(P))
=φN A
i( α(P A)) = φP A,α
i( ).
In he ollowing example we will show he cons uc ion o a ough Shapley unc ion,
which will be a coali ional Shapley unc ion di e en om he Owen alue and om
he classic Shapley unc ion.
Example. We aim o de ine a pa icula amily o ough coe icien s α. To his end,
we need a p e ious cons uc ion. Suppose ha Ais a ini e se and Sa nonemp y se
wi h S(A. Fo all T⊆Awe de ine
hA(S, T) = |S∩T|
|S|2|A|−1,
and hA(A, A) = hA(∅,∅) = 1. I we andomly (and wi h equal p obabili y) choose an
elemen in Sand hen andomly (and wi h equal p obabili y) choose an subse o A
ha con ains he elemen chosen, he numbe hA(S, T) is equal o he p obabili y ha
he subse o Achosen is equal o T. The ollowing equali y holds:
X
T∈[∅,A]
hA(S, T)=1.(11)
I |S|=s,|A|=aand p=|S∩T| hen
X
T∈[∅,A]
hA(S, T) = X
T∈[∅,A]
|S∩T|
|S|2|A|−1=1
|S|2|A|−1X
T∈[∅,A]|S∩T|
=1
s2a−1
s
X
p=0 s
p2a−sp=1
s2s−1
s
X
p=1 s
pp
=1
s2s−1
s
X
p=1 s−1
p−1s=1
2s−1
s−1
X
p=0 s−1
p= 1
whe e we ha e used wo well-known combina o ial iden i ies2. Now conside he pa -
i ion P={A1, ..., Am}. Fo each S⊆Nand T∈[˚
S, ¯
S], we de ine
αS
T=Y
k∈HP(S)
hAk(Sk, Tk),
2P
k=0 
k= 2 ,k
k=  −1
k−1
16

whe e Sk=S∩Akand HP(S) = {k:Sk6=∅}. Le us p o e ha αis a ough
coe icien .
Sha pness: since HP(Ap) = {p}we ha e ha αAp
Ap=hAp(Ap, Ap) = 1.
In a iance: suppose ha S⊆N Ap. Clea ly, HP(S∪Ap) = HP(S)∪{p}, (S∪Ap)p=
Apand (S∪A)k=Skwi h k6=p. Mo eo e , i T∈[˚
S, ¯
S] hen T∪Apsa is ies he
same p ope ies. We ha e
αS∪Ap
T∪Ap=hAp(Ap, Ap)Y
k∈HP(S)
hAk(Sk, Tk) = αS
T.
Comple eness: le S⊆N Ap,R∈[˚
S, ¯
S] and T∈(S, S ∪Ap). Le T=S∪K
wi h K∈(∅, Ap). Since HP(S∪K) = HP(S)∪ {p}we ha e ha , i k6=p, hen
(S∪K)k=Skand (R∪L)k=Rk. Besides, (S∪K)p=Kand (R∪L)p=L. We ha e
X
L∈[∅,Ap]
αS∪K
R∪L=X
L∈[∅,Ap]Y
k∈HP(S∪K)
hAk((S∪K)k,(R∪L)k)
=X
L∈[∅,Ap]
hAp(K, L)Y
k∈HP(S)
hAk(Sk, Rk)
=Y
k∈HP(S)
hAk(Sk, Rk)X
L∈[∅,Ap]
hAp(K, L)
=Y
k∈HP(S)
hAk(Sk, Rk) = αS
R=αS∪Ap
R∪Ap,
whe e we ha e used (11) and in a iance.
I we ake he coe icien abo e α(P) o each pa i ion Pwe ob ain a amily o ough
coe icien s, because, by cons uc ion, his amily sa is ies symme y and educ ion.
The e o e, φαis a ough Shapley unc ion, and, by Theo em 5.1, a coali ional Shapley
unc ion.
Le us calcula e φα( ) o a pa icula pa i ion Pand a pa icula game ∈G{1,2,3}.
Take P={{1,2},{3}} and he unanimi y game u{13}∈G{1,2,3}. We ha e ha
h{1,2}({1},∅) = h{1,2}({2},∅) = h{1,2}({1},{2}) = h{1,2}({2},{1}) = 0,
h{1,2}({1},{1}) = h{1,2}({2},{2}) = h{1,2}({1},{1,2}) = h{12}({2},{1,2})=1/2.
Besides, by de ini ion,
h{1,2}(∅,∅) = h{1,2}({1,2},{1,2}) = h{3}(∅,∅) = h{3}({3},{3})=1.
The ough e sion o a game ∈GNis
α({1}) = 1
2 ({1}) + 1
2 ({1,2}), α({2}) = 1
2 ({2}) + 1
2 ({1,2}), α({3}) = ({3}),
α({1,2}) = ({1,2}), α({1,3}) = 1
2 ({1,3})+1
2 (N), α({2,3}) = 1
2 ({2,3})+1
2 (N),
17
α(N) = (N).
In pa icula , uα
{1,3}({1,3}) = 1, uα
{1,3}({2,3})=1/2, uα
{1,3}(N) = 1 and uα
{1,3}(S)=0
o he wise. The classical Shapley alue and he Owen alue coincide o his game,
e u ning he payo ec o
φN(u{1,3}) = (1/2,0,1/2).
I is easy o check ha
φα(u{1,3}) = (4/12,1/12,7/12).
No ice ha φαdoes no sa is y he null playe p ope y. Indeed, φα
2(u{1,3})>0, al-
hough playe 2 is null in u{1,3}. This playe ecei es a payo o being pa o a
coali ion, {1,2}, ha is no i ele an .
The ollowing equali y is a classic exe cise o linea algeb a ha we will use in he
p oo o Theo em 5.2. Le x1, x2..., xn, y1, y2, ..., yn∈Rand n∈N. Then,
Dn=
x1+y1x1··· x1
x2x2+y2··· x2
.
.
..
.
.··· .
.
.
xnxn··· xn+yn

=
n
Y
k=1
yk"1 +
n
X
k=1
xk
yk#.(12)
Mo eo e , Dn=xk
n
Y
{l=1,l6=k}
yl+ykDk
n−1, whe e Dk
n−1is he de e minan ob ained when
we dele e ow kand column kin he exp ession o Dn
The las esul o his pape is de o ed o p o e ha he ough Shapley unc ions
“almos ” comple e he amily o coali ional Shapley unc ions. A pa i ion Pis said
o be a pa i ion in o la ge g oups i |A|>2 o e e y A∈ P. Games wi h a p io i
la ge unions model si ua ions which a e in e es ing o analyze. Fo ins ance, he Owen
alue was applied o hese games o s udy he powe in he Eu opean Pa liamen [6]
and in he In e na ional Mone a y Fund [16].
Theo em 5.2. I is a coali ional Shapley unc ion hen i coincides wi h a ough
Shapley unc ion on pa i ions in o la ge g oups.
P oo . Le be a coali ional Shapley unc ion. Le Pbe a pa i ion in o la ge g oups,
ha is, |A|>2 o all A∈ P. We will p o e he esul by induc ion on he ca dinali y
|P| o he pa i ion.
Base case, |P| = 1.Conside es ic ed o he games o e he globalis pa i ion
P={N}. No ice ha in his case compa ible symme y means symme y because all
he pe mu a ions o e Na e compa ible wi h {N}. Hence, is a alue ha sa is ies
e iciency, linea i y and symme y (an ELS- alue3 ollowing [18]). P oposi ion 2 in [18]
says ha he e is a coe icien bs o each ca dinali y s= 0, ..., n wi h b0= 14,bn= 1
3Mo eo e , he null g oup p ope y only says {N}(0) = 0. Bu combining e iciency wi h symme y he same
equali y is achie ed. Thus bo h amilies o alues can be iden i ied.
4Ac ually, in [18] ook b0= 0 bu his numbe is no signi ican .
18
and
{N}
i( ) = X
{S⊆N:i∈S}
γn
s[bs (S)−bs−1 (S {i})] .
We ake, o each non-c isp coali ion S,αS
T= 0 i ∅ 6=T6=S,αS
S=b|S|and αS
∅= 1 −
b|S|. Obse e ha we ha e cons uc ed a ough coe icien o pa i ion {N}. Sha pness:
he only c isp coali ions in his pa i ion a e Nand ∅, and bn=b0= 1. In a iance:
i is ue because bn=b0. Comple eness: le S=∅,A=Nand T∈2N {∅, N}. We
ha e
X
L∈[∅,N]
αT
L=αT
∅+b|T|= 1 = bn.
The e o e, using his ough coe icien , we ob ain φ{N},α = {N}.
Induc ion s ep. Suppose ha coincides wi h a ough Shapley alue o pa i ions
wi h ca dinali y less han m, ha is, he e exis s a amily o ough coe icien s αsuch
ha
φP,α = P(13)
o any pa i ion Pwi h |P| < m.
Take a pa i ion P={A1, ..., Am}. Le N=SA∈P A={1, ..., n}. We look o a ough
coe icien α(P) = α o de ine P=φP,α.
Reduc ion condi ion implies ha all he numbe s αS
Twi h S∩Ak=∅(o S∩Ak=Ak
by in a iance) o k= 1, ..., m a e de e mined by (13). Following Sec ion 3, coe icien
αmus sa is y he ough magic cube. By (10) all he numbe s αS
Twi h T∩Ak=∅ o
some g oup k= 1, ..., m a e de e mined. Hence, only he numbe s αS
Twi h T∩Ak6=∅
and S∩Ak6=Ak o all k= 1, ..., m a e no de e mined.
The ough coe icien mus sa is y he equa ions φP,α
i( ) = P
i( ) o each i∈Nand
game on N. I we use he linea i y axiom o we can educe he sys em o he
equa ions
φi(uα
H) = φP,α
i(uH) = P
i(uH) (14)
o all i∈Nand H⊆Nnonemp y. Mo eo e , we can assume H∩Ak6=∅ o
all k= 1, ..., m. Indeed, i H∩Ak=∅ o some k= 1, ..., m hen, by he null g oup
p ope y, P
i(uH) = P Ak
i(uH) o any i∈N Akand P
i(uH) = 0 i i∈Ak. The e o e
he payo s in he games a e de e mined. The same happens wi h any easible φP,α.
Suppose H=T1∪T2∪ ···Tmwi h Tk⊆Akand Tk6=∅ o any k= 1, ..., m. We
conside he se s Tkin H(and also in P) such ha :
•Tk=Ak o k= 1, ..., p (p∈ {0, ..., m}),
•Tk6=Akand |Tk|>1 o k=p+ 1, ..., p +q(q∈ {0, ..., m −p}), and
• |Tk|= 1 o k=p+q+ 1, ..., m.
Now we use he axiom o compa ible symme y o Pand φP,α o educe he numbe
o equali ies in (14). So, we ob ain he ollowing cases:
•Fo k= 1, ..., p : he e is only one di e en payo in Ak,φik(uα
H) = P
ik(uH),
because all he playe s a e in e nal symme ic playe s in uH.
19
•Fo k=p+ 1, ..., m : he e a e wo di e en payo s in Ak,φik(uα
H) = P
ik(uH)
and φjk(uα
H) = P
jk(uH) wi h ik∈Tkand jk∈Ak Tk, because all he playe s in
Tka e in e nal symme ic playe s in uHand also all he playe s in Ak Tk.
So, we ha e i e ypes o payo s (one in he i s kind o equa ions, wo in he second
one, and wo in he hi d one). We educe (14) o p+ 2(m−p) equa ions wi h he
s uc u e
X
{S⊆N:ik∈S}
γn
|S|[uα
H(S)−uα
H(S {ik})] = P
ik(uH), k = 1, ..., m (15)
X
{S⊆N:jk∈S}
γn
|S|[uα
H(S)−uα
H(S {jk})] = P
jk(uH), k =p+ 1, ..., m (16)
whe e
uα
H(S) = X
R∈[˚
S∪H, ¯
S]
αS
R.
We suppose hem in he ollowing o de : o each k= 1, ..., p he equa ion in (15) and
o e e y k=p+ 1, ..., m i s he equa ion in (15) and second he equa ion in (16).
We choose one playe ep esen ing each g oup, ha is, dH
k∈Ak o any k= 1, ..., m,
such ha dH
k∈Tk o all k= 1, ..., p +qand dH
k∈Ak Tk o all k=p+q+ 1, ..., m.
This elec ion is easible because he g oups a e la ge. Le
DH={dH
1, ..., dH
m}.
We selec p+ 2(m−p) unknown numbe s o sol e (15,16):
• o k= 1, ..., m :αDH∪{iH
k}
H, wi h iH
k∈Tk {dH
k} o equa ion (15),
• o k=p+ 1, ..., m :αDH∪{jH
k}
H, wi h jH
k∈Ak (Tk∪{dH
k}) o equa ion (16).
We conside hese unknowns o de ed like hei associa ed equa ions. Le ak=|Ak|
and k=|Tk| o k= 1, ..., m. We will ake in o accoun he symme y condi ion o
αin o de o join equal numbe s and ge he s uc u e o he ma ix in Figu e 2,
whe e we ocus on he coe icien s o ou unknown numbe s in o de o s udy hei
de e minan . The explana ion o how we ge he di e en coe icien s is in he Annex
o he pape . We no e ha some numbe s Xl
bwi h l= 1,2,3,4,5 appea in Figu e 2,
whose o mula ion can be seen in he Annex, which a e independen o he unknowns
and which a e epea ed in all he equa ions o each ow o blocks. In a second s ep
we ake ou Xq
bwi h q= 1, ..., 5 o he di e en ows, obse e ha Xq
b6= 0 o all
k= 1, ..., 5. We also ake ou he alues
ac−1
2, c−1
2, ac− c,ac−2
2
om he columns o kinds 2,3,4 and 6 espec i ely. They a e no ze o because he
g oups a e la ge. The eade mus ake in o accoun he elemen o he main diagonal
in each case. The esul is Figu e 3. We ans o m he main diagonal as
20
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p+1,...,p+q
<la exi sha1_base64="l Lj5ayaNSocN comVVIL7kzeBE=">AAAB6HicdZDLSgMxFIbPeK31VnXpJliELsowo8XaXcGNyw 2Am0pm RMG5u5kGSEM QddCXqzu xBXwb02kFF 1XX87/B85/ FhwpR3nw1pZXV 2Mx 5bd3d 2CweHLRUlkmGTRSKSHY8qFDzEpuZaYCeWSANPYNubXM399j1KxaPwVk9j7Ad0FHK M6 NqO2Wbdsux4NC0bF mcgCqpUl1Fzi2k6mIizVGBTee8OIJQGGmgmqVNd1Y 1PqdScCZzle4nCmLIJHWHXYEgDVP00W3dGT 1IEj1Gk 2/Z1MaKDUNPJMJqB6 3958+J XTbR/2U95GCcaQ2Yix MTQXRE5q3JkE kWkwNUCa52ZKwMZWUaXOb Kn/1ZH8D60z272wz28qxXppeYgcHMMJlMCFK ThGh QBAYTeIQXeLXu A yXpeRFes5Z8j+CH 7RMXgIxZ</la exi >
1,...,p
<la exi sha1_base64="nmhA55uQ6b0yLqaOCK /43IzGW0=">AAAB5HicdZDNTgIxFIX 4B/iH+ STSMxYTWZUSKyI3HjEqMICUxIp9yBhs5P2o4JIbyB oy684l8Ad/GMoyJGj2 /ecJ dcPxFcac 5sAo q2 G8XN0 b2zu5ee /gTsWpZNhmsYhl16cKBY+w bkW2E0k0 AX2PEnlwu/c49S8Ti61dME ZCOIh5wR UZ3 CBPyhXHLuRiSyhXsuh4RLXdjJVIFd UH7 D2OWhhhpJqhSPddJ DejUnMmcF7qpwoTyiZ0hD2DEQ1RebNs1Tk5CWJJ9BhJ9 6endFQqWnom0xI9Vj99hbD 7xeqoMLb8ajJNUYMRMxXpAKomOyaEyGXCLTYmqAMsnNloSNqaRMm7uUTP2 juR/uDu13XP77LpWaVbzQxThCI6hCi7UoQlX0II2MBjBI7zAqxVYD9aT9byMFqz8zyH8kPX2C xli9g=</la exi >
ib
<la exi sha1_base64="5Fkg35Z2FUnas1FHsX 3l mo68=">AAAB5HicdZDLTgIxFIbP4A3xh p000hMWJEZJSI7EjcuMcolgQnplDNQ6VzSdkwI4Q10ZdSdT+QL+DaWYUzU6L/6e 6/y mPFwuu G1/WLmV1bX1j xmYW 7Z3e uH/QVlEiGbZYJCLZ9ahCwUNsaa4FdmOJNPAEd zJ5cL 3KNUPAp 9TRGN6CjkPucUW1GN3cDb1As2ZV6K KEWjWDukOcip2qBJmag+J7 xixJMBQM0GV6jl2 N0ZlZozg NCP1EYUzahI+wZDGmAyp2lq87JiR9JosdI0 37IwGSk0Dz2QCqs q 7cY/uX1Eu1 uDMexonGkJmI8 xEEB2RRWMy5BKZFlMDlElu iRsTCVl2 ylYOp/dST/Q/u04pxXzq6 pUY5O0QejuAYyuBADRpwBU1oAYMRPMIL Fq+9WA9Wc/LaM7K/hzCD1l n/3li9k=</la exi >
jb
<la exi sha1_base64="R +Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/W 6pLN8EiFFqGGS3W7gpuXFaw TAdSibN KGZSUwyQhn6F oSde T+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXV aO4Wd a3 ndK+8 9BRPJCZdzBmX wFShNGYdDXVjNwKSVAUMNIPppcL 39PpKI8 EzQ wIjWMaUoy0GXmi5 Z 266L2 2wXHHsVia4hGYjh5YLXd JVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyC WEwOz9PZuiSKlZFJhMhPRE/ YWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qn n un103Ku1q ogiOALHoApc0AR cAU6oAsw4OAR IBXK7Ye C eRk WPm Q/BD1 snhHiNuA==</la exi >
p+1,...,p+q
<la exi sha1_base64="Pd5UqyzbYAsz 0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/W 6pLN8EiFFqGGS3W7gpuXFaw TAdSibN KHJZEwyQhn6F oSde T+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyu a+kZxs7S1 bO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQ TC8X +eSEVFdKNnM E5Gkc0pBhpM/Li2l3N du2Xe DcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpS Zi P4UkoJNQTA P392yKuFIzHpgMR3qi nuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qn n un103Ku1q ogiOALHoApc0AR cAU6oAswEOAR IBXK7Ie C eRk WPm Q/BD1 sngYKN Q==</la exi >
p+q+1,...,m
<la exi sha1_base64="nmhA55uQ6b0yLqaOCK /43IzGW0=">AAAB5HicdZDNTgIxFIX 4B/iH+ STSMxYTWZUSKyI3HjEqMICUxIp9yBhs5P2o4JIbyB oy684l8Ad/GMoyJGj2 /ecJ dcPxFcac 5sAo q2 G8XN0 b2zu5ee /gTsWpZNhmsYhl16cKBY+w bkW2E0k0 AX2PEnlwu/c49S8Ti61dME ZCOIh5wR UZ3 CBPyhXHLuRiSyhXsuh4RLXdjJVIFd UH7 D2OWhhhpJqhSPddJ DejUnMmcF7qpwoTyiZ0hD2DEQ1RebNs1Tk5CWJJ9BhJ9 6endFQqWnom0xI9Vj99hbD 7xeqoMLb8ajJNUYMRMxXpAKomOyaEyGXCLTYmqAMsnNloSNqaRMm7uUTP2 juR/uDu13XP77LpWaVbzQxThCI6hCi7UoQlX0II2MBjBI7zAqxVYD9aT9byMFqz8zyH8kPX2C xli9g=</la exi >
ib
<la exi sha1_base64="Pd5UqyzbYAsz 0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/W 6pLN8EiFFqGGS3W7gpuXFaw TAdSibN KHJZEwyQhn6F oSde T+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyu a+kZxs7S1 bO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQ TC8X +eSEVFdKNnM E5Gkc0pBhpM/Li2l3N du2Xe DcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpS Zi P4UkoJNQTA P392yKuFIzHpgMR3qi nuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qn n un103Ku1q ogiOALHoApc0AR cAU6oAswEOAR IBXK7Ie C eRk WPm Q/BD1 sngYKN Q==</la exi >
p+q+1,...,m
<la exi sha1_base64="5Fkg35Z2FUnas1FHsX 3l mo68=">AAAB5HicdZDLTgIxFIbP4A3xh p000hMWJEZJSI7EjcuMcolgQnplDNQ6VzSdkwI4Q10ZdSdT+QL+DaWYUzU6L/6e 6/y mPFwuu G1/WLmV1bX1j xmYW 7Z3e uH/QVlEiGbZYJCLZ9ahCwUNsaa4FdmOJNPAEd zJ5cL 3KNUPAp 9TRGN6CjkPucUW1GN3cDb1As2ZV6K KEWjWDukOcip2qBJmag+J7 xixJMBQM0GV6jl2 N0ZlZozg NCP1EYUzahI+wZDGmAyp2lq87JiR9JosdI0 37IwGSk0Dz2QCqs q 7cY/uX1Eu1 uDMexonGkJmI8 xEEB2RRWMy5BKZFlMDlElu iRsTCVl2 ylYOp/dST/Q/u04pxXzq6 pUY5O0QejuAYyuBADRpwBU1oAYMRPMIL Fq+9WA9Wc/LaM7K/hzCD1l n/3li9k=</la exi >
jb
<la exi sha1_base64="4nlCEpLs9l 42ASaJ36EZgRcakA=">AAAB4nicdZDNTgIxFIX 4B/iH+ STSMxYTWZUSKyI3HjEhIBE5iQT kDDZ2 B0TMuEFdGXUnY/kC/g2FhgTNXpWX+85Te65 iK40o7zYRXW1jc2 4 bpZ3d 2D8uFRV8WpZNhhsYjlnU8VCh5hR3M 8C6RSEN YM+ Xi/83j1KxePoVs8S9EI6jnjAGdVm1PaH5YpjN5YiK6jXcmi4xLWdpSqQqzUs w9GMU DjDQTVKm+6yTay6jUnAmclwapwoSyKR1j32BEQ1Re lx0Ts6CWBI9QbJ8 89mNFRqF omE1I9Ub+9x A 5/q4M LeJSkGiNmIsYLUkF0TBZ9yYhLZF MDFAmudmSsAmVlGlzlZKp/9WR/A/dc9u9 C/a Uqzmh+iCCdwClVwoQ5NuIEWdIABwiO8wKs1sh6sJ+ 5FS1Y+Z9j+CH 7ROG94 8</la exi >
b
<la exi sha1_base64="ihMRwXZ4j cYe29bFDJV6p2VUNM=">AAAB7nicdZDNSsNAFIUn9a/W 6pLN4NF6CokVazdFdy4ES YH2hDmUx 2qGTSZyZCKX0NXQl6s6H8QV8GydpBBU9qzP3nIH7XT/mTGnH+bAKK6 6x FzdLW9s7uXnn/oKOiRFJo04hHsucTBZwJaGumO RiCST0OXT96WWad+9BKhaJWz2LwQ JWLCAUaLNyLsmTOARI+NIED4sVxy7kQk T 0sNw0Xu7aTqYJy Ybl98EookkIQlNOlOq7Tqy9OZGaUQ6L0iBREBM6JWPoGy ICMqbZ0s 8EkQSawngLP39+6chE NQ 90QqIn6neWD /K+okOL w5E3GiQVBTMVmQcKwjnLIbUAlU81lKTCUzW2I6IZJQbS5UM h jPh/06nZ7 l9elO NK 5IY oCB2jKnJRHTXRFWqhNqLoDj2iF/RqxdaD9WQ9L6sFK/9ziH7Ie sEF8KP9A==</la exi >
Main diagonal
<la exi sha1_base64="4nlCEpLs9l 42ASaJ36EZgRcakA=">AAAB4nicdZDNTgIxFIX 4B/iH+ STSMxYTWZUSKyI3HjEhIBE5iQT kDDZ2 B0TMuEFdGXUnY/kC/g2FhgTNXpWX+85Te65 iK40o7zYRXW1jc2 4 bpZ3d 2D8uFRV8WpZNhhsYjlnU8VCh5hR3M 8C6RSEN YM+ Xi/83j1KxePoVs8S9EI6jnjAGdVm1PaH5YpjN5YiK6jXcmi4xLWdpSqQqzUs w9GMU DjDQTVKm+6yTay6jUnAmclwapwoSyKR1j32BEQ1Re lx0Ts6CWBI9QbJ8 89mNFRqF omE1I9Ub+9x A 5/q4M LeJSkGiNmIsYLUkF0TBZ9yYhLZF MDFAmudmSsAmVlGlzlZKp/9WR/A/dc9u9 C/a Uqzmh+iCCdwClVwoQ5NuIEWdIABwiO8wKs1sh6sJ+ 5FS1Y+Z9j+CH 7ROG94 8</la exi >
b
<la exi sha1_base64=" JDXT3 CNBUD89M9a9S2FUm509c=">AAAB/nicdZDLSsNAGIUn9Vb Leqym8EidBUSLdbuC osoK hSYNk+mkHTq5MDMRSgjoy+hK1J0P4Q 4Nk7TCCp6V /85wz85/diRoU0zQ+ LK6 5R3qxsbe/s7un7B30RJRyTHo5YxAceEoTRkPQklYwMYk5Q4DFy480uF 7NLeGCRuG1nM ECdAkpD7FSKqRq1d xOIpGqWXo46Nk9hOqYsVZpnbc WaabRywSU0GwW0LGgZZq4aKNR19Xd7HOEkIKHEDAkx MxYOinikmJGsoqdCBIjPEMTMlQYooAIJ81LZPDYjziUUwLz9/dsigIh5oGnMgGSU/HbWwz/8oaJ9M+dlIZxIkmIVUR5 sKgjODiFnBMOcGSzRUgzKnaEuIp4ghLdbGKq /VE 4P/RPDOjNO xq1d 04RBlUwRGoAws0QR 0QB 0AAb34BG8gF Tn Qn TnZbSkFX8OwQ9pb59HcJY9</la exi >
↵DH[{iH
c}
H
<la exi sha1_base64="l Lj5ayaNSocN comVVIL7kzeBE=">AAAB6HicdZDLSgMxFIbPeK31VnXpJliELsowo8XaXcGNyw 2Am0pm RMG5u5kGSEM QddCXqzu xBXwb02kFF 1XX87/B85/ FhwpR3nw1pZXV 2Mx 5bd3d 2CweHLRUlkmGTRSKSHY8qFDzEpuZaYCeWSANPYNubXM399j1KxaPwVk9j7Ad0FHK M6 NqO2Wbdsux4NC0bF mcgCqpUl1Fzi2k6mIizVGBTee8OIJQGGmgmqVNd1Y 1PqdScCZzle4nCmLIJHWHXYEgDVP00W3dGT 1IEj1Gk 2/Z1MaKDUNPJMJqB6 3958+J XTbR/2U95GCcaQ2Yix MTQXRE5q3JkE kWkwNUCa52ZKwMZWUaXOb Kn/1ZH8D60z272wz28qxXppeYgcHMMJlMCFK ThGh QBAYTeIQXeLXu A yXpeRFes5Z8j+CH 7RMXgIxZ</la exi >
1,...,p
<la exi sha1_base64="R +Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/W 6pLN8EiFFqGGS3W7gpuXFaw TAdSibN KGZSUwyQhn6F oSde T+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXV aO4Wd a3 ndK+8 9BRPJCZdzBmX wFShNGYdDXVjNwKSVAUMNIPppcL 39PpKI8 EzQ wIjWMaUoy0GXmi5 Z 266L2 2wXHHsVia4hGYjh5YLXd JVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyC WEwOz9PZuiSKlZFJhMhPRE/ YWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qn n un103Ku1q ogiOALHoApc0AR cAU6oAsw4OAR IBXK7Ye C eRk WPm Q/BD1 snhHiNuA==</la exi >
p+1,...,p+q
<la exi sha1_base64="R +Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/W 6pLN8EiFFqGGS3W7gpuXFaw TAdSibN KGZSUwyQhn6F oSde T+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXV aO4Wd a3 ndK+8 9BRPJCZdzBmX wFShNGYdDXVjNwKSVAUMNIPppcL 39PpKI8 EzQ wIjWMaUoy0GXmi5 Z 266L2 2wXHHsVia4hGYjh5YLXd JVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyC WEwOz9PZuiSKlZFJhMhPRE/ YWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qn n un103Ku1q ogiOALHoApc0AR cAU6oAsw4OAR IBXK7Ye C eRk WPm Q/BD1 snhHiNuA==</la exi >
p+1,...,p+q
<la exi sha1_base64="Pd5UqyzbYAsz 0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/W 6pLN8EiFFqGGS3W7gpuXFaw TAdSibN KHJZEwyQhn6F oSde T+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyu a+kZxs7S1 bO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQ TC8X +eSEVFdKNnM E5Gkc0pBhpM/Li2l3N du2Xe DcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpS Zi P4UkoJNQTA P392yKuFIzHpgMR3qi nuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qn n un103Ku1q ogiOALHoApc0AR cAU6oAswEOAR IBXK7Ie C eRk WPm Q/BD1 sngYKN Q==</la exi >
p+q+1,...,m
<la exi sha1_base64="Pd5UqyzbYAsz 0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/W 6pLN8EiFFqGGS3W7gpuXFaw TAdSibN KHJZEwyQhn6F oSde T+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyu a+kZxs7S1 bO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQ TC8X +eSEVFdKNnM E5Gkc0pBhpM/Li2l3N du2Xe DcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpS Zi P4UkoJNQTA P392yKuFIzHpgMR3qi nuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qn n un103Ku1q ogiOALHoApc0AR cAU6oAswEOAR IBXK7Ie C eRk WPm Q/BD1 sngYKN Q==</la exi >
p+q+1,...,m
<la exi sha1_base64=" JDXT3 CNBUD89M9a9S2FUm509c=">AAAB/nicdZDLSsNAGIUn9Vb Leqym8EidBUSLdbuC osoK hSYNk+mkHTq5MDMRSgjoy+hK1J0P4Q 4Nk7TCCp6V /85wz85/diRoU0zQ+ LK6 5R3qxsbe/s7un7B30RJRyTHo5YxAceEoTRkPQklYwMYk5Q4DFy480uF 7NLeGCRuG1nM ECdAkpD7FSKqRq1d xOIpGqWXo46Nk9hOqYsVZpnbc WaabRywSU0GwW0LGgZZq4aKNR19Xd7HOEkIKHEDAkx MxYOinikmJGsoqdCBIjPEMTMlQYooAIJ81LZPDYjziUUwLz9/dsigIh5oGnMgGSU/HbWwz/8oaJ9M+dlIZxIkmIVUR5 sKgjODiFnBMOcGSzRUgzKnaEuIp4ghLdbGKq /VE 4P/RPDOjNO xq1d 04RBlUwRGoAws0QR 0QB 0AAb34BG8gF Tn Qn TnZbSkFX8OwQ9pb59HcJY9</la exi >
↵DH[{iH
c}
H
<la exi sha1_base64=" JDXT3 CNBUD89M9a9S2FUm509c=">AAAB/nicdZDLSsNAGIUn9Vb Leqym8EidBUSLdbuC osoK hSYNk+mkHTq5MDMRSgjoy+hK1J0P4Q 4Nk7TCCp6V /85wz85/diRoU0zQ+ LK6 5R3qxsbe/s7un7B30RJRyTHo5YxAceEoTRkPQklYwMYk5Q4DFy480uF 7NLeGCRuG1nM ECdAkpD7FSKqRq1d xOIpGqWXo46Nk9hOqYsVZpnbc WaabRywSU0GwW0LGgZZq4aKNR19Xd7HOEkIKHEDAkx MxYOinikmJGsoqdCBIjPEMTMlQYooAIJ81LZPDYjziUUwLz9/dsigIh5oGnMgGSU/HbWwz/8oaJ9M+dlIZxIkmIVUR5 sKgjODiFnBMOcGSzRUgzKnaEuIp4ghLdbGKq /VE 4P/RPDOjNO xq1d 04RBlUwRGoAws0QR 0QB 0AAb34BG8gF Tn Qn TnZbSkFX8OwQ9pb59HcJY9</la exi >
↵DH[{iH
c}
H
<la exi sha1_base64="2 4dygS7EDahRMOK CxEREiuqo=">AAAB/nicdZDLSsNAGIUn9Vb Leqym8EidBUSLdbuC osoK9QJOGyXTSjp1cmJkIJQT0ZXQl6s6H8AV8G6dpBRU9q2/+cwb+83sxo0Ka5odWWFldW98obpa2 nd29/T9g66IEo5JB0cs4n0PCcJoSDqSSkb6MSco8BjpedOLud+7JVzQKLyWs5g4ARqH1KcYSTVy9bKNWDxBw/Ry2LJxE pjYsVZpnbc WKaTRywQXUa0 oWNAyzFwVsFTb1d/ UYSTgIQSMyTEwDJj6aSIS4oZyUp2IkiM8BSNyUBhiAIinDQ kcFjP+JQTgjM39+zKQqEmAWeygRITsR bz78yxsk0j93Uh GiSQhVhHl+QmDMoLzW8AR5QRLNlOAMKdqS4gniCMs1cVKq 5XR/g/dE8M68w4 apVm XlIYqgDI5AFVigDpqgBdqgAzC4B4/gBbxqd9qD9qQ9L6IFb nnEPyQ9 YJSPeWPg==</la exi >
↵DH[{jH
c}
H
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X3
b
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X3
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X3
b
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X3
b
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X4
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X4
b
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X4
b
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X4
b
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X4
b
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X4
b
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X5
b
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X5
b
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X5
b
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X5
b
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X5
b
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X5
b
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ac1
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ac1
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ac1
2
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ab2
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c1
2
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c1
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c1
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ac2
2
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ac2
2
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ac2
2
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(ac c)
<la exi sha1_base64="OQNxgDjd3+ahFYdQ1ZY+iAKZ g0=">AAAB6nicbZC9TsMwFIV +C3l 8DIYlEhlYEqAQSMlVgYi0R/UB Vjn TW WTyHaQqqg ARMCNh6HF+B cEsGaDnT53uOpX ukAiuje +OU LK6 64WN4ubW9s5uaW+/qeNUMWywWMSqHVCNgk YMNwIbCcKqQwE oLRzdR PaLSPI7uzThBX9JBxEPOqLGjhw sVPTYye9U m ujORR ByKEOueq/02e3HLJUYGSao1h3PTYy UWU4EzgpdlONCWUjOsCOxYhK1H42W3hCjsNYETNEMn /zmZUaj2Wgc1IaoZ63psO//M6qQm /YxHSWowYjZi TAVxMRk2p 0uUJmxNgCZY bLQkbUkWZsdcp2 e NlFaJ5V c q+d1FuVbJD1GAQziCCnhwBTW4hTo0gIGEZ3iDd0c4T86L8/oTXXLyPw wR87HN5J5jS4=</la exi >
(ac c)
<la exi sha1_base64="OQNxgDjd3+ahFYdQ1ZY+iAKZ g0=">AAAB6nicbZC9TsMwFIV +C3l 8DIYlEhlYEqAQSMlVgYi0R/UB Vjn TW WTyHaQqqg ARMCNh6HF+B cEsGaDnT53uOpX ukAiuje +OU LK6 64WN4ubW9s5uaW+/qeNUMWywWMSqHVCNgk YMNwIbCcKqQwE oLRzdR PaLSPI7uzThBX9JBxEPOqLGjhw sVPTYye9U m ujORR ByKEOueq/02e3HLJUYGSao1h3PTYy UWU4EzgpdlONCWUjOsCOxYhK1H42W3hCjsNYETNEMn /zmZUaj2Wgc1IaoZ63psO//M6qQm /YxHSWowYjZi TAVxMRk2p 0uUJmxNgCZY bLQkbUkWZsdcp2 e NlFaJ5V c q+d1FuVbJD1GAQziCCnhwBTW4hTo0gIGEZ3iDd0c4T86L8/oTXXLyPw wR87HN5J5jS4=</la exi >
(ac c)
<la exi sha1_base64="Qks+Fj9l0Nj56B Dyn7mdLogLE0=">AAACCnicbVDNSsNAGNzU 1 /oh69LBaxIi1JFPVY8OKxg 2BNIbNd Mu3U3C7kYooW+gL6MnUW8e QH xm0NqK1zmp2ZhW8mSBiVy I+jcLC4 LySnG1 La+sbllbu+0ZJwKTJo4Z HoBEgSRiPSVFQx0kkEQTxgpB0MLyd++44ISePoRo0S4nHUj2hIMVJa8s1D1+n2Ee Iz/ixPb6NqhXlB1Xn6Ed1 O 5Z mqWVPAeWLnpAxyNHzzo9uLccpJpDBDU q2lSg Q0JRzMi41E0lSRAeoj5xNY0QJ9LLpoXG8CCMBVQDAq 39kMcSlHPNAZj RAznoT8T/PTVV44WU0SlJFIqwj2g TBlUMJ7 AHhUEKzbSBGFB9ZUQD5BAWOn1S q+PV 2n Scmn1WO7k+Ldc +RBFsA 2QQXY4BzUwRVogCbA4AE8gV wZ wbj8az8 IdLRj5n13wB8b7F16kmUc=</la exi >
[2n
m+1 ( b2)n
m+2]
<la exi sha1_base64="oAEos6TWFNW9EimHi PNRhuPnGU=">AAAB/nicbVDLSsNAFJ34 PVVddlNsAgFoSQq6 LgxmUF+4Cmlsn0Jhk6k4SZG6GEg 6M kTd+RH+gH/j Hah Wd17j3nwj3HTwXX6Dh 1 Lyyu aemGjuLm1 bNb2 6SRTDJosEYnq+FSD4DE0kaOATqqASl9A2x9eT T2PSjNk/gWRyn0JA1jHnBG0az6pbInIMCuF1IpaT+Xx+74L YUDyOs9UsVp+ZMYS8Sd0YqZIZG /TpDRKWSYiRCap113VS7OVUIWcCxkU 05BSNqQhdA2NqQTdy6chx ZRkCgbI7Cn829 TqXWI+kbj6QY6Xl s xP62YYXPZyHqcZQsyMxWhBJmxM7EkX9oA YChGhlCmuPnSZhFVlKFp Gjiu/NhF0n pOae105 zi 16qyIAimTQ1IlL kgdXJNGqRJGHkkz+SN FsP1pP1Y 3+WJes2c0B+QP 4x Q+JWR</la exi >
⇥n
m+1
<la exi sha1_base64="pa7 IYcBZN R9xgYMV0huw mcxU=">AAACCnicbVDLSsNAFJ34 6qL 0Ei1gRS6KiLgU3LhWsCk0MN+NNOziThJkboYT+g 6M kTdu QH/Bun Q By6ce88ZmHPiXApDn hjIyOjU9MTk1XZmbn5heqi0 nJis0xybPZKY YzAoRYpNEiTxM cIKpZ4Ed8c9 WLW9RGZOkZdXMMFbRTkQgOZE9RdT2QmFBjqw5R EV2/I2gDUpBVK N7d5VGmjR7lAYVW ewx A/U 8IamxIU6i6n wn FCYUpcgjE 38spLEGT4BJ7laAwmAO/gTa2LE1BoQnLQaCeu5Zk2qUOuoP9u7cEZUxXxdajgD m 9Y//qe1CkoOwlKkeUGYcmuxWlJIlzK334 7LTRykl1LgG h +nyDmjgZNu 2Pj+77B/y l2w99 7Jzu1g7 wyKm2ApbZXXms312yI7ZCWsyzu7ZI3 h 86d8+A8Oc9 1hFn+GaZ/YDz9gmO Zls</la exi >
(ab b1)n
m+2⇤
<la exi sha1_base64="qW nJlH+cj gVO 6knB7eb 8HlA=">AAAB73icbZDLSgMxFIYz9Vb e SzWARCkKZUVGXBTcuK9gL9MaZ9EwbmmTGJCOUoc+hK1F3 os 4NuY1llo67/6c 4/cP4TxJxp43l Tm5ldW19I79Z2N e2d0 7h80dJQoinUa8Ui1A DImcS6YYZjK1YIIuDYDMY3M7/5iEqzSN6bSYxdAUPJQkbB2FG MwQhoJ+KU3/ak/1iya 4c7nL4GdQIplq/eJnZxDRRKA0lIPWbd+LTTcFZRjlOC10Eo0x0DEMsW1RgkDdTedbT92TMFKuGaE7 //OpiC0nojAZgSYkV70ZsP/ HZiwu uymScGJTURqwXJ w1kTs 7w6YQm 4xAJQxeyWLh2BAm siQq2 9YdhkaZxX/snJ+d1Gql ND5MkROSZl4pM UiW3pEbqhBJFnskbeXcenC nxXn9ieac7M8h+SPn4xsMUo/U</la exi >
n
m+1
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[2n
m+1 (ab3)n
m+2]
<la exi sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPU Oj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEV TCDiRXjbpM7IOwlhYqk+sy 3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAA PV5B13dmy86TZ XinFVO k+L dJ0iDzaQ/uohBx0jm oC VRA2H0gJ7QK3oz7o1H49l4+Y7mjOm X QHx sXx7mY9w==</la exi >
[n
m+1 (ab1)n
m+2]
<la exi sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPU Oj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEV TCDiRXjbpM7IOwlhYqk+sy 3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAA PV5B13dmy86TZ XinFVO k+L dJ0iDzaQ/uohBx0jm oC VRA2H0gJ7QK3oz7o1H49l4+Y7mjOm X QHx sXx7mY9w==</la exi >
[n
m+1 (ab1)n
m+2]
<la exi sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPU Oj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEV TCDiRXjbpM7IOwlhYqk+sy 3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAA PV5B13dmy86TZ XinFVO k+L dJ0iDzaQ/uohBx0jm oC VRA2H0gJ7QK3oz7o1H49l4+Y7mjOm X QHx sXx7mY9w==</la exi >
[n
m+1 (ab1)n
m+2]
<la exi sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPU Oj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEV TCDiRXjbpM7IOwlhYqk+sy 3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAA PV5B13dmy86TZ XinFVO k+L dJ0iDzaQ/uohBx0jm oC VRA2H0gJ7QK3oz7o1H49l4+Y7mjOm X QHx sXx7mY9w==</la exi >
[n
m+1 (ab1)n
m+2]
<la exi sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPU Oj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEV TCDiRXjbpM7IOwlhYqk+sy 3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAA PV5B13dmy86TZ XinFVO k+L dJ0iDzaQ/uohBx0jm oC VRA2H0gJ7QK3oz7o1H49l4+Y7mjOm X QHx sXx7mY9w==</la exi >
[n
m+1 (ab1)n
m+2]
<la exi sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzW 1 /Vj16CRalIi27VdRjwY HC YH uuSTbN aJJdkqxQlj6B oyeRL159QV8G9NaUF nNJmZwDcTJowq7Ti Vm5hcWl5Jb9aWF 2Nyy 3eaKk4lJg0cs1i2Q6QIo4I0NNWM BNJEA8ZaYWDy7H uiNS0Vjc6GFC I56gkYUI22kwD70Oj3EOQoy uyObkW5pIOw7B79qFWj+oFddC OBHCeuFNSBFPUA/uj041xyonQmCGlPNdJ J8hqSlmZFTopIokCA9Qj3iGCsSJ8 NJnxE8iGIJdZ/Ay 3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQl RcCXE SYS1Ga9g6 uzZedJs1pxzyon16 FWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9 Edz1 TPL gD6/0L5Y+ZCg==</la exi >
[n
m+1 ( b1)n
m+2]
<la exi sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzW 1 /Vj16CRalIi27VdRjwY HC YH uuSTbN aJJdkqxQlj6B oyeRL159QV8G9NaUF nNJmZwDcTJowq7Ti Vm5hcWl5Jb9aWF 2Nyy 3eaKk4lJg0cs1i2Q6QIo4I0NNWM BNJEA8ZaYWDy7H uiNS0Vjc6GFC I56gkYUI22kwD70Oj3EOQoy uyObkW5pIOw7B79qFWj+oFddC OBHCeuFNSBFPUA/uj041xyonQmCGlPNdJ J8hqSlmZFTopIokCA9Qj3iGCsSJ8 NJnxE8iGIJdZ/Ay 3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQl RcCXE SYS1Ga9g6 uzZedJs1pxzyon16 FWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9 Edz1 TPL gD6/0L5Y+ZCg==</la exi >
[n
m+1 ( b1)n
m+2]
<la exi sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzW 1 /Vj16CRalIi27VdRjwY HC YH uuSTbN aJJdkqxQlj6B oyeRL159QV8G9NaUF nNJmZwDcTJowq7Ti Vm5hcWl5Jb9aWF 2Nyy 3eaKk4lJg0cs1i2Q6QIo4I0NNWM BNJEA8ZaYWDy7H uiNS0Vjc6GFC I56gkYUI22kwD70Oj3EOQoy uyObkW5pIOw7B79qFWj+oFddC OBHCeuFNSBFPUA/uj041xyonQmCGlPNdJ J8hqSlmZFTopIokCA9Qj3iGCsSJ8 NJnxE8iGIJdZ/Ay 3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQl RcCXE SYS1Ga9g6 uzZedJs1pxzyon16 FWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9 Edz1 TPL gD6/0L5Y+ZCg==</la exi >
[n
m+1 ( b1)n
m+2]
<la exi sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzW 1 /Vj16CRalIi27VdRjwY HC YH uuSTbN aJJdkqxQlj6B oyeRL159QV8G9NaUF nNJmZwDcTJowq7Ti Vm5hcWl5Jb9aWF 2Nyy 3eaKk4lJg0cs1i2Q6QIo4I0NNWM BNJEA8ZaYWDy7H uiNS0Vjc6GFC I56gkYUI22kwD70Oj3EOQoy uyObkW5pIOw7B79qFWj+oFddC OBHCeuFNSBFPUA/uj041xyonQmCGlPNdJ J8hqSlmZFTopIokCA9Qj3iGCsSJ8 NJnxE8iGIJdZ/Ay 3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQl RcCXE SYS1Ga9g6 uzZedJs1pxzyon16 FWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9 Edz1 TPL gD6/0L5Y+ZCg==</la exi >
[n
m+1 ( b1)n
m+2]
<la exi sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzW 1 /Vj16CRalIi27VdRjwY HC YH uuSTbN aJJdkqxQlj6B oyeRL159QV8G9NaUF nNJmZwDcTJowq7Ti Vm5hcWl5Jb9aWF 2Nyy 3eaKk4lJg0cs1i2Q6QIo4I0NNWM BNJEA8ZaYWDy7H uiNS0Vjc6GFC I56gkYUI22kwD70Oj3EOQoy uyObkW5pIOw7B79qFWj+oFddC OBHCeuFNSBFPUA/uj041xyonQmCGlPNdJ J8hqSlmZFTopIokCA9Qj3iGCsSJ8 NJnxE8iGIJdZ/Ay 3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQl RcCXE SYS1Ga9g6 uzZedJs1pxzyon16 FWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9 Edz1 TPL gD6/0L5Y+ZCg==</la exi >
[n
m+1 ( b1)n
m+2]
<la exi sha1_base64="N/+1 M9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZM O7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1 57Z3d UJx/6BpokQz3mCRjHQ7AMOlCHkDBU ejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0E VaXX6YJWSW3HnoM Ey0iJZKj3ip/d sQSxUNkEozpeG6M goaBZN8mu8mhs AxjDkHU DUNz46 zwKT0ZRJ iiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRn Wn aE5QzmxBJgW9k KRqCBo 1S3 b3Fssuk2a14l1Wzu8uS Vy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenB n9Se64mQ7h+QPnI9 72KQPQ==</la exi >
n
m+2
<la exi sha1_base64="N/+1 M9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZM O7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1 57Z3d UJx/6BpokQz3mCRjHQ7AMOlCHkDBU ejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0E VaXX6YJWSW3HnoM Ey0iJZKj3ip/d sQSxUNkEozpeG6M goaBZN8mu8mhs AxjDkHU DUNz46 zwKT0ZRJ iiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRn Wn aE5QzmxBJgW9k KRqCBo 1S3 b3Fssuk2a14l1Wzu8uS Vy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenB n9Se64mQ7h+QPnI9 72KQPQ==</la exi >
n
m+2
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m+2
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m+2
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m+2
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m+2
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m+2
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m+2
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m+2
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m+2
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m+1 (ab2)n
m+2]
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m+1 (ab2)n
m+2]
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m+1 (ab2)n
m+2]
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Figu e 2. Kinds o ows and main diagonal. S ep 1.
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p+1,...,p+q
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1,...,p
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ib
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jb
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p+1,...,p+q
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p+q+1,...,m
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ib
<la exi sha1_base64="Pd5UqyzbYAsz 0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/W 6pLN8EiFFqGGS3W7gpuXFaw TAdSibN KHJZEwyQhn6F oSde T+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyu a+kZxs7S1 bO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQ TC8X +eSEVFdKNnM E5Gkc0pBhpM/Li2l3N du2Xe DcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpS Zi P4UkoJNQTA P392yKuFIzHpgMR3qi nuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qn n un103Ku1q ogiOALHoApc0AR cAU6oAswEOAR IBXK7Ie C eRk WPm Q/BD1 sngYKN Q==</la exi >
p+q+1,...,m
<la exi sha1_base64="5Fkg35Z2FUnas1FHsX 3l mo68=">AAAB5HicdZDLTgIxFIbP4A3xh p000hMWJEZJSI7EjcuMcolgQnplDNQ6VzSdkwI4Q10ZdSdT+QL+DaWYUzU6L/6e 6/y mPFwuu G1/WLmV1bX1j xmYW 7Z3e uH/QVlEiGbZYJCLZ9ahCwUNsaa4FdmOJNPAEd zJ5cL 3KNUPAp 9TRGN6CjkPucUW1GN3cDb1As2ZV6K KEWjWDukOcip2qBJmag+J7 xixJMBQM0GV6jl2 N0ZlZozg NCP1EYUzahI+wZDGmAyp2lq87JiR9JosdI0 37IwGSk0Dz2QCqs q 7cY/uX1Eu1 uDMexonGkJmI8 xEEB2RRWMy5BKZFlMDlElu iRsTCVl2 ylYOp/dST/Q/u04pxXzq6 pUY5O0QejuAYyuBADRpwBU1oAYMRPMIL Fq+9WA9Wc/LaM7K/hzCD1l n/3li9k=</la exi >
jb
<la exi sha1_base64="4nlCEpLs9l 42ASaJ36EZgRcakA=">AAAB4nicdZDNTgIxFIX 4B/iH+ STSMxYTWZUSKyI3HjEhIBE5iQT kDDZ2 B0TMuEFdGXUnY/kC/g2FhgTNXpWX+85Te65 iK40o7zYRXW1jc2 4 bpZ3d 2D8uFRV8WpZNhhsYjlnU8VCh5hR3M 8C6RSEN YM+ Xi/83j1KxePoVs8S9EI6jnjAGdVm1PaH5YpjN5YiK6jXcmi4xLWdpSqQqzUs w9GMU DjDQTVKm+6yTay6jUnAmclwapwoSyKR1j32BEQ1Re lx0Ts6CWBI9QbJ8 89mNFRqF omE1I9Ub+9x A 5/q4M LeJSkGiNmIsYLUkF0TBZ9yYhLZF MDFAmudmSsAmVlGlzlZKp/9WR/A/dc9u9 C/a Uqzmh+iCCdwClVwoQ5NuIEWdIABwiO8wKs1sh6sJ+ 5FS1Y+Z9j+CH 7ROG94 8</la exi >
b
<la exi sha1_base64="ihMRwXZ4j cYe29bFDJV6p2VUNM=">AAAB7nicdZDNSsNAFIUn9a/W 6pLN4NF6CokVazdFdy4ES YH2hDmUx 2qGTSZyZCKX0NXQl6s6H8QV8GydpBBU9qzP3nIH7XT/mTGnH+bAKK6 6x FzdLW9s7uXnn/oKOiRFJo04hHsucTBZwJaGumO RiCST0OXT96WWad+9BKhaJWz2LwQ JWLCAUaLNyLsmTOARI+NIED4sVxy7kQk T 0sNw0Xu7aTqYJy Ybl98EookkIQlNOlOq7Tqy9OZGaUQ6L0iBREBM6JWPoGy ICMqbZ0s 8EkQSawngLP39+6chE NQ 90QqIn6neWD /K+okOL w5E3GiQVBTMVmQcKwjnLIbUAlU81lKTCUzW2I6IZJQbS5UM h jPh/06nZ7 l9elO NK 5IY oCB2jKnJRHTXRFWqhNqLoDj2iF/RqxdaD9WQ9L6sFK/9ziH7Ie sEF8KP9A==</la exi >
Main diagonal
<la exi sha1_base64="4nlCEpLs9l 42ASaJ36EZgRcakA=">AAAB4nicdZDNTgIxFIX 4B/iH+ STSMxYTWZUSKyI3HjEhIBE5iQT kDDZ2 B0TMuEFdGXUnY/kC/g2FhgTNXpWX+85Te65 iK40o7zYRXW1jc2 4 bpZ3d 2D8uFRV8WpZNhhsYjlnU8VCh5hR3M 8C6RSEN YM+ Xi/83j1KxePoVs8S9EI6jnjAGdVm1PaH5YpjN5YiK6jXcmi4xLWdpSqQqzUs w9GMU DjDQTVKm+6yTay6jUnAmclwapwoSyKR1j32BEQ1Re lx0Ts6CWBI9QbJ8 89mNFRqF omE1I9Ub+9x A 5/q4M LeJSkGiNmIsYLUkF0TBZ9yYhLZF MDFAmudmSsAmVlGlzlZKp/9WR/A/dc9u9 C/a Uqzmh+iCCdwClVwoQ5NuIEWdIABwiO8wKs1sh6sJ+ 5FS1Y+Z9j+CH 7ROG94 8</la exi >
b
<la exi sha1_base64=" JDXT3 CNBUD89M9a9S2FUm509c=">AAAB/nicdZDLSsNAGIUn9Vb Leqym8EidBUSLdbuC osoK hSYNk+mkHTq5MDMRSgjoy+hK1J0P4Q 4Nk7TCCp6V /85wz85/diRoU0zQ+ LK6 5R3qxsbe/s7un7B30RJRyTHo5YxAceEoTRkPQklYwMYk5Q4DFy480uF 7NLeGCRuG1nM ECdAkpD7FSKqRq1d xOIpGqWXo46Nk9hOqYsVZpnbc WaabRywSU0GwW0LGgZZq4aKNR19Xd7HOEkIKHEDAkx MxYOinikmJGsoqdCBIjPEMTMlQYooAIJ81LZPDYjziUUwLz9/dsigIh5oGnMgGSU/HbWwz/8oaJ9M+dlIZxIkmIVUR5 sKgjODiFnBMOcGSzRUgzKnaEuIp4ghLdbGKq /VE 4P/RPDOjNO xq1d 04RBlUwRGoAws0QR 0QB 0AAb34BG8gF Tn Qn TnZbSkFX8OwQ9pb59HcJY9</la exi >
↵DH[{iH
c}
H
<la exi sha1_base64="l Lj5ayaNSocN comVVIL7kzeBE=">AAAB6HicdZDLSgMxFIbPeK31VnXpJliELsowo8XaXcGNyw 2Am0pm RMG5u5kGSEM QddCXqzu xBXwb02kFF 1XX87/B85/ FhwpR3nw1pZXV 2Mx 5bd3d 2CweHLRUlkmGTRSKSHY8qFDzEpuZaYCeWSANPYNubXM399j1KxaPwVk9j7Ad0FHK M6 NqO2Wbdsux4NC0bF mcgCqpUl1Fzi2k6mIizVGBTee8OIJQGGmgmqVNd1Y 1PqdScCZzle4nCmLIJHWHXYEgDVP00W3dGT 1IEj1Gk 2/Z1MaKDUNPJMJqB6 3958+J XTbR/2U95GCcaQ2Yix MTQXRE5q3JkE kWkwNUCa52ZKwMZWUaXOb Kn/1ZH8D60z272wz28qxXppeYgcHMMJlMCFK ThGh QBAYTeIQXeLXu A yXpeRFes5Z8j+CH 7RMXgIxZ</la exi >
1,...,p
<la exi sha1_base64="R +Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/W 6pLN8EiFFqGGS3W7gpuXFaw TAdSibN KGZSUwyQhn6F oSde T+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXV aO4Wd a3 ndK+8 9BRPJCZdzBmX wFShNGYdDXVjNwKSVAUMNIPppcL 39PpKI8 EzQ wIjWMaUoy0GXmi5 Z 266L2 2wXHHsVia4hGYjh5YLXd JVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyC WEwOz9PZuiSKlZFJhMhPRE/ YWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qn n un103Ku1q ogiOALHoApc0AR cAU6oAsw4OAR IBXK7Ye C eRk WPm Q/BD1 snhHiNuA==</la exi >
p+1,...,p+q
<la exi sha1_base64="R +Lo7KN1sZ8SxN10zu0qijMDlY=">AAAB7HicdZDNSgMxFIUz9a/W 6pLN8EiFFqGGS3W7gpuXFaw TAdSibN KGZSUwyQhn6F oSde T+AK+jel0BBU9qy/3nMA9NxCMKu04H1ZhZXV aO4Wd a3 ndK+8 9BRPJCZdzBmX wFShNGYdDXVjNwKSVAUMNIPppcL 39PpKI8 EzQ wIjWMaUoy0GXmi5 Z 266L2 2wXHHsVia4hGYjh5YLXd JVAG5OsPy+2DEcRKRWGOGlPJcR2g/RVJTzMi8NEgUEQhP0Zh4BmMUEeWn2cpzeBJyC WEwOz9PZuiSKlZFJhMhPRE/ YWw788L9HhhZ/SWCSaxNhEjBcmDGoOF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qn n un103Ku1q ogiOALHoApc0AR cAU6oAsw4OAR IBXK7Ye C eRk WPm Q/BD1 snhHiNuA==</la exi >
p+1,...,p+q
<la exi sha1_base64="Pd5UqyzbYAsz 0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/W 6pLN8EiFFqGGS3W7gpuXFaw TAdSibN KHJZEwyQhn6F oSde T+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyu a+kZxs7S1 bO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQ TC8X +eSEVFdKNnM E5Gkc0pBhpM/Li2l3N du2Xe DcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpS Zi P4UkoJNQTA P392yKuFIzHpgMR3qi nuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qn n un103Ku1q ogiOALHoApc0AR cAU6oAswEOAR IBXK7Ie C eRk WPm Q/BD1 sngYKN Q==</la exi >
p+q+1,...,m
<la exi sha1_base64="Pd5UqyzbYAsz 0/kj3Go7gN/yk8=">AAAB7HicdZDNSgMxFIUz9a/W 6pLN8EiFFqGGS3W7gpuXFaw TAdSibN KHJZEwyQhn6F oSde T+AK+jel0BBU9qy/3nMA9N4gZVdpxPqzCyu a+kZxs7S1 bO7V94/6CmRSEy6WDAhbwOkCKMR6WqqGbmNJUE8YKQ TC8X +eSEVFdKNnM E5Gkc0pBhpM/Li2l3N du2Xe DcsWxW5ngEpqNHFoudG0nUwXk6gzL74ORwAknkcYMKeW5Tqz9FElNMSPz0iBRJEZ4isbEMxghTpS Zi P4UkoJNQTA P392yKuFIzHpgMR3qi nuL4V+el+jwwk9pFCeaRNhEjBcmDGoBF83hiEqCNZsZQFhSsyXEEyQR1uY+JVP/qyP8H3qn n un103Ku1q ogiOALHoApc0AR cAU6oAswEOAR IBXK7Ie C eRk WPm Q/BD1 sngYKN Q==</la exi >
p+q+1,...,m
<la exi sha1_base64=" JDXT3 CNBUD89M9a9S2FUm509c=">AAAB/nicdZDLSsNAGIUn9Vb Leqym8EidBUSLdbuC osoK hSYNk+mkHTq5MDMRSgjoy+hK1J0P4Q 4Nk7TCCp6V /85wz85/diRoU0zQ+ LK6 5R3qxsbe/s7un7B30RJRyTHo5YxAceEoTRkPQklYwMYk5Q4DFy480uF 7NLeGCRuG1nM ECdAkpD7FSKqRq1d xOIpGqWXo46Nk9hOqYsVZpnbc WaabRywSU0GwW0LGgZZq4aKNR19Xd7HOEkIKHEDAkx MxYOinikmJGsoqdCBIjPEMTMlQYooAIJ81LZPDYjziUUwLz9/dsigIh5oGnMgGSU/HbWwz/8oaJ9M+dlIZxIkmIVUR5 sKgjODiFnBMOcGSzRUgzKnaEuIp4ghLdbGKq /VE 4P/RPDOjNO xq1d 04RBlUwRGoAws0QR 0QB 0AAb34BG8gF Tn Qn TnZbSkFX8OwQ9pb59HcJY9</la exi >
↵DH[{iH
c}
H
<la exi sha1_base64=" JDXT3 CNBUD89M9a9S2FUm509c=">AAAB/nicdZDLSsNAGIUn9Vb Leqym8EidBUSLdbuC osoK hSYNk+mkHTq5MDMRSgjoy+hK1J0P4Q 4Nk7TCCp6V /85wz85/diRoU0zQ+ LK6 5R3qxsbe/s7un7B30RJRyTHo5YxAceEoTRkPQklYwMYk5Q4DFy480uF 7NLeGCRuG1nM ECdAkpD7FSKqRq1d xOIpGqWXo46Nk9hOqYsVZpnbc WaabRywSU0GwW0LGgZZq4aKNR19Xd7HOEkIKHEDAkx MxYOinikmJGsoqdCBIjPEMTMlQYooAIJ81LZPDYjziUUwLz9/dsigIh5oGnMgGSU/HbWwz/8oaJ9M+dlIZxIkmIVUR5 sKgjODiFnBMOcGSzRUgzKnaEuIp4ghLdbGKq /VE 4P/RPDOjNO xq1d 04RBlUwRGoAws0QR 0QB 0AAb34BG8gF Tn Qn TnZbSkFX8OwQ9pb59HcJY9</la exi >
↵DH[{iH
c}
H
<la exi sha1_base64="2 4dygS7EDahRMOK CxEREiuqo=">AAAB/nicdZDLSsNAGIUn9Vb Leqym8EidBUSLdbuC osoK9QJOGyXTSjp1cmJkIJQT0ZXQl6s6H8AV8G6dpBRU9q2/+cwb+83sxo0Ka5odWWFldW98obpa2 nd29/T9g66IEo5JB0cs4n0PCcJoSDqSSkb6MSco8BjpedOLud+7JVzQKLyWs5g4ARqH1KcYSTVy9bKNWDxBw/Ry2LJxE pjYsVZpnbc WKaTRywQXUa0 oWNAyzFwVsFTb1d/ UYSTgIQSMyTEwDJj6aSIS4oZyUp2IkiM8BSNyUBhiAIinDQ kcFjP+JQTgjM39+zKQqEmAWeygRITsR bz78yxsk0j93Uh GiSQhVhHl+QmDMoLzW8AR5QRLNlOAMKdqS4gniCMs1cVKq 5XR/g/dE8M68w4 apVm XlIYqgDI5AFVigDpqgBdqgAzC4B4/gBbxqd9qD9qQ9L6IFb nnEPyQ9 YJSPeWPg==</la exi >
↵DH[{jH
c}
H
<la exi sha1_base64="2 4dygS7EDahRMOK CxEREiuqo=">AAAB/nicdZDLSsNAGIUn9Vb Leqym8EidBUSLdbuC osoK9QJOGyXTSjp1cmJkIJQT0ZXQl6s6H8AV8G6dpBRU9q2/+cwb+83sxo0Ka5odWWFldW98obpa2 nd29/T9g66IEo5JB0cs4n0PCcJoSDqSSkb6MSco8BjpedOLud+7JVzQKLyWs5g4ARqH1KcYSTVy9bKNWDxBw/Ry2LJxE pjYsVZpnbc WKaTRywQXUa0 oWNAyzFwVsFTb1d/ UYSTgIQSMyTEwDJj6aSIS4oZyUp2IkiM8BSNyUBhiAIinDQ kcFjP+JQTgjM39+zKQqEmAWeygRITsR bz78yxsk0j93Uh GiSQhVhHl+QmDMoLzW8AR5QRLNlOAMKdqS4gniCMs1cVKq 5XR/g/dE8M68w4 apVm XlIYqgDI5AFVigDpqgBdqgAzC4B4/gBbxqd9qD9qQ9L6IFb nnEPyQ9 YJSPeWPg==</la exi >
↵DH[{jH
c}
H
<la exi sha1_base64="oWnImmMjVj3ALKw/FQW5HpDcQ5k=">AAACCnicbVDNSsNAGNzU 1 /qh69BI YkZYkinosePFYw 5AGsOX7bZdupuE3Y1QQ 9AX0ZPo 48+gK+jdsaUF nNDszC99MEDMqlWV9G mFxaXll xqYW19Y3O uL3TlFEiMGngiEWiHYAkjIakoahipB0LAjxgpBUMLyd+644ISaPwRo1i4nHoh7RHMSg +cVD1+n0gXPwU35sj2/DShn8oOIc/aiOVj2/WLKq1hTmPLEzUkIZ6n7xo9ONcMJJqDADKV3bipWXglAUMzIudBJJYsBD6BNX0xA4kV46LTQ2D3qRMNWAmNP372wKXMoRD3SGgx IWW8i/ue5iepdeCkN40SREOuI9noJM1VkTnYxu1QQ NhIE8CC6i NPAABWOn1C q+PV 2njSdqn1WPbk+LdXK2RB5 I 2URnZ6BzV0BWqowbC6AE9oV 0Z wbj8az8 IdzRnZn130B8b7F0DOmTQ=</la exi >
[2n
m+1 (ab2)n
m+2]
<la exi sha1_base64="Qks+Fj9l0Nj56B Dyn7mdLogLE0=">AAACCnicbVDNSsNAGNzU 1 /oh69LBaxIi1JFPVY8OKxg 2BNIbNd Mu3U3C7kYooW+gL6MnUW8e QH xm0NqK1zmp2ZhW8mSBiVy I+jcLC4 LySnG1 La+sbllbu+0ZJwKTJo4Z HoBEgSRiPSVFQx0kkEQTxgpB0MLyd++44ISePoRo0S4nHUj2hIMVJa8s1D1+n2Ee Iz/ixPb6NqhXlB1Xn6Ed1 O 5Z mqWVPAeWLnpAxyNHzzo9uLccpJpDBDU q2lSg Q0JRzMi41E0lSRAeoj5xNY0QJ9LLpoXG8CCMBVQDAq 39kMcSlHPNAZj RAznoT8T/PTVV44WU0SlJFIqwj2g TBlUMJ7 AHhUEKzbSBGFB9ZUQD5BAWOn1S q+PV 2n Scmn1WO7k+Ldc +RBFsA 2QQXY4BzUwRVogCbA4AE8gV wZ wbj8az8 IdLRj5n13wB8b7F16kmUc=</la exi >
[2n
m+1 ( b2)n
m+2]
<la exi sha1_base64="oAEos6TWFNW9EimHi PNRhuPnGU=">AAAB/nicbVDLSsNAFJ34 PVVddlNsAgFoSQq6 LgxmUF+4Cmlsn0Jhk6k4SZG6GEg 6M kTd+RH+gH/j Hah Wd17j3nwj3HTwXX6Dh 1 Lyyu aemGjuLm1 bNb2 6SRTDJosEYnq+FSD4DE0kaOATqqASl9A2x9eT T2PSjNk/gWRyn0JA1jHnBG0az6pbInIMCuF1IpaT+Xx+74L YUDyOs9UsVp+ZMYS8Sd0YqZIZG /TpDRKWSYiRCap113VS7OVUIWcCxkU 05BSNqQhdA2NqQTdy6chx ZRkCgbI7Cn829 TqXWI+kbj6QY6Xl s xP62YYXPZyHqcZQsyMxWhBJmxM7EkX9oA YChGhlCmuPnSZhFVlKFp Gjiu/NhF0n pOae105 zi 16qyIAimTQ1IlL kgdXJNGqRJGHkkz+SN FsP1pP1Y 3+WJes2c0B+QP 4x Q+JWR</la exi >
⇥n
m+1
<la exi sha1_base64="pa7 IYcBZN R9xgYMV0huw mcxU=">AAACCnicbVDLSsNAFJ34 6qL 0Ei1gRS6KiLgU3LhWsCk0MN+NNOziThJkboYT+g 6M kTdu QH/Bun Q By6ce88ZmHPiXApDn hjIyOjU9MTk1XZmbn5heqi0 nJis0xybPZKY YzAoRYpNEiTxM cIKpZ4Ed8c9 WLW9RGZOkZdXMMFbRTkQgOZE9RdT2QmFBjqw5R EV2/I2gDUpBVK N7d5VGmjR7lAYVW ewx A/U 8IamxIU6i6n wn FCYUpcgjE 38spLEGT4BJ7laAwmAO/gTa2LE1BoQnLQaCeu5Zk2qUOuoP9u7cEZUxXxdajgD m 9Y//qe1CkoOwlKkeUGYcmuxWlJIlzK334 7LTRykl1LgG h +nyDmjgZNu 2Pj+77B/y l2w99 7Jzu1g7 wyKm2ApbZXXms312yI7ZCWsyzu7ZI3 h 86d8+A8Oc9 1hFn+GaZ/YDz9gmO Zls</la exi >
(ab b1)n
m+2⇤
<la exi sha1_base64="qW nJlH+cj gVO 6knB7eb 8HlA=">AAAB73icbZDLSgMxFIYz9Vb e SzWARCkKZUVGXBTcuK9gL9MaZ9EwbmmTGJCOUoc+hK1F3 os 4NuY1llo67/6c 4/cP4TxJxp43l Tm5ldW19I79Z2N e2d0 7h80dJQoinUa8Ui1A DImcS6YYZjK1YIIuDYDMY3M7/5iEqzSN6bSYxdAUPJQkbB2FG MwQhoJ+KU3/ak/1iya 4c7nL4GdQIplq/eJnZxDRRKA0lIPWbd+LTTcFZRjlOC10Eo0x0DEMsW1RgkDdTedbT92TMFKuGaE7 //OpiC0nojAZgSYkV70ZsP/ HZiwu uymScGJTURqwXJ w1kTs 7w6YQm 4xAJQxeyWLh2BAm siQq2 9YdhkaZxX/snJ+d1Gql ND5MkROSZl4pM UiW3pEbqhBJFnskbeXcenC nxXn9ieac7M8h+SPn4xsMUo/U</la exi >
n
m+1
<la exi sha1_base64="ibUg aD1 EWb628nJkJLDqkWyug=">AAACCnicbVDNSsNAGNzU 1 /oh69BI YkZakF VY8OKxg 2BNoY 2027dDcJuxuhhL6B oyeRL159AV8G7c1oFbnNDszC9+MHzMqlW1/GLmFxaXll xqYW19Y3PL3N5pySgRmDRxxCLR8UESRkPSVFQx0okFAe4z0 ZHF1O/ UuEpFF4 cYxcTkMQhpQDEpLnnnY YGwDl4KT92JjdhuQSeX64d a V bqeWbQ 9gzWX+JkpIgyNDzz dePcMJJqDADKbuOHSs3BaEoZmRS6CWSxIBHMCBdTUPgRL p NDEOggiYakhsWb n9kUuJRj7usMBzWU895U/M/ Jio4d1MaxokiIdYR7QUJs1RkTXex+lQQ NhYE8CC6is PAQBWOn1C q+M1/2L2lVK85ppXZ1UqyXsiHyaA/ oxJy0Bmqo0 UQE2E0T16RC/o1bgzHown4/k mjOyP7 oF4y3T0JcmTU=</la exi >
[2n
m+1 (ab3)n
m+2]
<la exi sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPU Oj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEV TCDiRXjbpM7IOwlhYqk+sy 3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAA PV5B13dmy86TZ XinFVO k+L dJ0iDzaQ/uohBx0jm oC VRA2H0gJ7QK3oz7o1H49l4+Y7mjOm X QHx sXx7mY9w==</la exi >
[n
m+1 (ab1)n
m+2]
<la exi sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPU Oj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEV TCDiRXjbpM7IOwlhYqk+sy 3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAA PV5B13dmy86TZ XinFVO k+L dJ0iDzaQ/uohBx0jm oC VRA2H0gJ7QK3oz7o1H49l4+Y7mjOm X QHx sXx7mY9w==</la exi >
[n
m+1 (ab1)n
m+2]
<la exi sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPU Oj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEV TCDiRXjbpM7IOwlhYqk+sy 3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAA PV5B13dmy86TZ XinFVO k+L dJ0iDzaQ/uohBx0jm oC VRA2H0gJ7QK3oz7o1H49l4+Y7mjOm X QHx sXx7mY9w==</la exi >
[n
m+1 (ab1)n
m+2]
<la exi sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPU Oj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEV TCDiRXjbpM7IOwlhYqk+sy 3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAA PV5B13dmy86TZ XinFVO k+L dJ0iDzaQ/uohBx0jm oC VRA2H0gJ7QK3oz7o1H49l4+Y7mjOm X QHx sXx7mY9w==</la exi >
[n
m+1 (ab1)n
m+2]
<la exi sha1_base64="TqUGq/co0ig8W3AoQe4a9Q0ObRU=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Qdo5+1KpWPd8s2hV7AmueOFNSRFPU Oj041xykmkMAMpXcdOlJeBUBQzMip0UkkSwAPoEV TCDiRXjbpM7IOwlhYqk+sy 3NgMu5ZAHOsNB9eWsNxb/89xUhRdeRqMkVSTCOqK9MGWWiq3xLFaXCoIVG2oCWFB9pYX7IAA PV5B13dmy86TZ XinFVO k+L dJ0iDzaQ/uohBx0jm oC VRA2H0gJ7QK3oz7o1H49l4+Y7mjOm X QHx sXx7mY9w==</la exi >
[n
m+1 (ab1)n
m+2]
<la exi sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzW 1 /Vj16CRalIi27VdRjwY HC YH uuSTbN aJJdkqxQlj6B oyeRL159QV8G9NaUF nNJmZwDcTJowq7Ti Vm5hcWl5Jb9aWF 2Nyy 3eaKk4lJg0cs1i2Q6QIo4I0NNWM BNJEA8ZaYWDy7H uiNS0Vjc6GFC I56gkYUI22kwD70Oj3EOQoy uyObkW5pIOw7B79qFWj+oFddC OBHCeuFNSBFPUA/uj041xyonQmCGlPNdJ J8hqSlmZFTopIokCA9Qj3iGCsSJ8 NJnxE8iGIJdZ/Ay 3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQl RcCXE SYS1Ga9g6 uzZedJs1pxzyon16 FWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9 Edz1 TPL gD6/0L5Y+ZCg==</la exi >
[n
m+1 ( b1)n
m+2]
<la exi sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzW 1 /Vj16CRalIi27VdRjwY HC YH uuSTbN aJJdkqxQlj6B oyeRL159QV8G9NaUF nNJmZwDcTJowq7Ti Vm5hcWl5Jb9aWF 2Nyy 3eaKk4lJg0cs1i2Q6QIo4I0NNWM BNJEA8ZaYWDy7H uiNS0Vjc6GFC I56gkYUI22kwD70Oj3EOQoy uyObkW5pIOw7B79qFWj+oFddC OBHCeuFNSBFPUA/uj041xyonQmCGlPNdJ J8hqSlmZFTopIokCA9Qj3iGCsSJ8 NJnxE8iGIJdZ/Ay 3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQl RcCXE SYS1Ga9g6 uzZedJs1pxzyon16 FWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9 Edz1 TPL gD6/0L5Y+ZCg==</la exi >
[n
m+1 ( b1)n
m+2]
<la exi sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzW 1 /Vj16CRalIi27VdRjwY HC YH uuSTbN aJJdkqxQlj6B oyeRL159QV8G9NaUF nNJmZwDcTJowq7Ti Vm5hcWl5Jb9aWF 2Nyy 3eaKk4lJg0cs1i2Q6QIo4I0NNWM BNJEA8ZaYWDy7H uiNS0Vjc6GFC I56gkYUI22kwD70Oj3EOQoy uyObkW5pIOw7B79qFWj+oFddC OBHCeuFNSBFPUA/uj041xyonQmCGlPNdJ J8hqSlmZFTopIokCA9Qj3iGCsSJ8 NJnxE8iGIJdZ/Ay 3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQl RcCXE SYS1Ga9g6 uzZedJs1pxzyon16 FWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9 Edz1 TPL gD6/0L5Y+ZCg==</la exi >
[n
m+1 ( b1)n
m+2]
<la exi sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzW 1 /Vj16CRalIi27VdRjwY HC YH uuSTbN aJJdkqxQlj6B oyeRL159QV8G9NaUF nNJmZwDcTJowq7Ti Vm5hcWl5Jb9aWF 2Nyy 3eaKk4lJg0cs1i2Q6QIo4I0NNWM BNJEA8ZaYWDy7H uiNS0Vjc6GFC I56gkYUI22kwD70Oj3EOQoy uyObkW5pIOw7B79qFWj+oFddC OBHCeuFNSBFPUA/uj041xyonQmCGlPNdJ J8hqSlmZFTopIokCA9Qj3iGCsSJ8 NJnxE8iGIJdZ/Ay 3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQl RcCXE SYS1Ga9g6 uzZedJs1pxzyon16 FWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9 Edz1 TPL gD6/0L5Y+ZCg==</la exi >
[n
m+1 ( b1)n
m+2]
<la exi sha1_base64="uh/jl0mxhsG+SCNs1somCZNjRKw=">AAACCXicbVDNSgMxGMzW 1 /Vj16CRalIi27VdRjwY HC YH uuSTbN aJJdkqxQlj6B oyeRL159QV8G9NaUF nNJmZwDcTJowq7Ti Vm5hcWl5Jb9aWF 2Nyy 3eaKk4lJg0cs1i2Q6QIo4I0NNWM BNJEA8ZaYWDy7H uiNS0Vjc6GFC I56gkYUI22kwD70Oj3EOQoy uyObkW5pIOw7B79qFWj+oFddC OBHCeuFNSBFPUA/uj041xyonQmCGlPNdJ J8hqSlmZFTopIokCA9Qj3iGCsSJ8 NJnxE8iGIJdZ/Ay 3NkNcqSEPTYYj3Vez3lj8z/NSHV34GRVJqonAJmK8KGVQx3A8C+xSSbBmQ0MQl RcCXE SYS1Ga9g6 uzZedJs1pxzyon16 FWmk6RB7sgX1QAi44BzVwBeqgATB4AE/gFbxZ99aj9Wy9 Edz1 TPL gD6/0L5Y+ZCg==</la exi >
[n
m+1 ( b1)n
m+2]
<la exi sha1_base64="N/+1 M9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZM O7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1 57Z3d UJx/6BpokQz3mCRjHQ7AMOlCHkDBU ejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0E VaXX6YJWSW3HnoM Ey0iJZKj3ip/d sQSxUNkEozpeG6M goaBZN8mu8mhs AxjDkHU DUNz46 zwKT0ZRJ iiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRn Wn aE5QzmxBJgW9k KRqCBo 1S3 b3Fssuk2a14l1Wzu8uS Vy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenB n9Se64mQ7h+QPnI9 72KQPQ==</la exi >
n
m+2
<la exi sha1_base64="N/+1 M9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZM O7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1 57Z3d UJx/6BpokQz3mCRjHQ7AMOlCHkDBU ejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0E VaXX6YJWSW3HnoM Ey0iJZKj3ip/d sQSxUNkEozpeG6M goaBZN8mu8mhs AxjDkHU DUNz46 zwKT0ZRJ iiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRn Wn aE5QzmxBJgW9k KRqCBo 1S3 b3Fssuk2a14l1Wzu8uS Vy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenB n9Se64mQ7h+QPnI9 72KQPQ==</la exi >
n
m+2
<la exi sha1_base64="N/+1 M9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZM O7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1 57Z3d UJx/6BpokQz3mCRjHQ7AMOlCHkDBU ejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0E VaXX6YJWSW3HnoM Ey0iJZKj3ip/d sQSxUNkEozpeG6M goaBZN8mu8mhs AxjDkHU DUNz46 zwKT0ZRJ iiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRn Wn aE5QzmxBJgW9k KRqCBo 1S3 b3Fssuk2a14l1Wzu8uS Vy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenB n9Se64mQ7h+QPnI9 72KQPQ==</la exi >
n
m+2
<la exi sha1_base64="N/+1 M9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZM O7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1 57Z3d UJx/6BpokQz3mCRjHQ7AMOlCHkDBU ejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0E VaXX6YJWSW3HnoM Ey0iJZKj3ip/d sQSxUNkEozpeG6M goaBZN8mu8mhs AxjDkHU DUNz46 zwKT0ZRJ iiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRn Wn aE5QzmxBJgW9k KRqCBo 1S3 b3Fssuk2a14l1Wzu8uS Vy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenB n9Se64mQ7h+QPnI9 72KQPQ==</la exi >
n
m+2
<la exi sha1_base64="N/+1 M9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZM O7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1 57Z3d UJx/6BpokQz3mCRjHQ7AMOlCHkDBU ejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0E VaXX6YJWSW3HnoM Ey0iJZKj3ip/d sQSxUNkEozpeG6M goaBZN8mu8mhs AxjDkHU DUNz46 zwKT0ZRJ iiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRn Wn aE5QzmxBJgW9k KRqCBo 1S3 b3Fssuk2a14l1Wzu8uS Vy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenB n9Se64mQ7h+QPnI9 72KQPQ==</la exi >
n
m+2
<la exi sha1_base64="N/+1 M9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZM O7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1 57Z3d UJx/6BpokQz3mCRjHQ7AMOlCHkDBU ejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0E VaXX6YJWSW3HnoM Ey0iJZKj3ip/d sQSxUNkEozpeG6M goaBZN8mu8mhs AxjDkHU DUNz46 zwKT0ZRJ iiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRn Wn aE5QzmxBJgW9k KRqCBo 1S3 b3Fssuk2a14l1Wzu8uS Vy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenB n9Se64mQ7h+QPnI9 72KQPQ==</la exi >
n
m+2
<la exi sha1_base64="N/+1 M9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZM O7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1 57Z3d UJx/6BpokQz3mCRjHQ7AMOlCHkDBU ejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0E VaXX6YJWSW3HnoM Ey0iJZKj3ip/d sQSxUNkEozpeG6M goaBZN8mu8mhs AxjDkHU DUNz46 zwKT0ZRJ iiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRn Wn aE5QzmxBJgW9k KRqCBo 1S3 b3Fssuk2a14l1Wzu8uS Vy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenB n9Se64mQ7h+QPnI9 72KQPQ==</la exi >
n
m+2
<la exi sha1_base64="N/+1 M9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZM O7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1 57Z3d UJx/6BpokQz3mCRjHQ7AMOlCHkDBU ejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0E VaXX6YJWSW3HnoM Ey0iJZKj3ip/d sQSxUNkEozpeG6M goaBZN8mu8mhs AxjDkHU DUNz46 zwKT0ZRJ iiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRn Wn aE5QzmxBJgW9k KRqCBo 1S3 b3Fssuk2a14l1Wzu8uS Vy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenB n9Se64mQ7h+QPnI9 72KQPQ==</la exi >
n
m+2
<la exi sha1_base64="N/+1 M9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZM O7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1 57Z3d UJx/6BpokQz3mCRjHQ7AMOlCHkDBU ejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0E VaXX6YJWSW3HnoM Ey0iJZKj3ip/d sQSxUNkEozpeG6M goaBZN8mu8mhs AxjDkHU DUNz46 zwKT0ZRJ iiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRn Wn aE5QzmxBJgW9k KRqCBo 1S3 b3Fssuk2a14l1Wzu8uS Vy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenB n9Se64mQ7h+QPnI9 72KQPQ==</la exi >
n
m+2
<la exi sha1_base64="N/+1 M9EXHjHMU74mYKsxEsNAWU=">AAAB8nicbVDLSgNBEJz1GeMjUY9eBoMQEMNuFPUY8OIxgnlAsobeySQZM O7zPQKYcmP6EnUm5/iD/g3TuIeNLFO1VXV0NVBLIVB1/1yVlbX1jc2c1 57Z3d UJx/6BpokQz3mCRjHQ7AMOlCHkDBU ejjUHFUjeCsY3M7/1yLURUXiPk5j7CoahGAgGaKVesXDWHYJS0E VaXX6YJWSW3HnoM Ey0iJZKj3ip/d sQSxUNkEozpeG6M goaBZN8mu8mhs AxjDkHU DUNz46 zwKT0ZRJ iiNP5/DubgjJmogKbUYAjs+jNxP+8ToKDaz8VYZwgD5mNWG+QSIoRn Wn aE5QzmxBJgW9k KRqCBo 1S3 b3Fssuk2a14l1Wzu8uS Vy9ogcOSLHpEw8ckVq5JbUSYMwkpBn8kbeHXSenB n9Se64mQ7h+QPnI9 72KQPQ==</la exi >
n
m+2
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[n
m+1 (ab2)n
m+2]
<la exi sha1_base64="Db51iQ8zL//n Js7 Xn1X3T/ps=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Q h79qFW e 5Z C 2BNY8caakiKao++ZHpx jlJNIYQZSuo6dKC8DoShmZFTopJIkgA QI66mEXAi WzSZ2QdhLGwVJ9Yk/ bAZcyiEPdIaD6s Zbyz+57mpCi+8jEZJqkiEdUR7YcosFV jWawuFQQ N QEsKD6Sg 3QQBWe yC u/Mlp0nzW FOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPx Px8h3NGdM/u+gPjPc yUeY+A==</la exi >
[n
m+1 (ab2)n
m+2]
<la exi sha1_base64="Db51iQ8zL//n Js7 Xn1X3T/ps=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Q h79qFW e 5Z C 2BNY8caakiKao++ZHpx jlJNIYQZSuo6dKC8DoShmZFTopJIkgA QI66mEXAi WzSZ2QdhLGwVJ9Yk/ bAZcyiEPdIaD6s Zbyz+57mpCi+8jEZJqkiEdUR7YcosFV jWawuFQQ N QEsKD6Sg 3QQBWe yC u/Mlp0nzW FOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPx Px8h3NGdM/u+gPjPc yUeY+A==</la exi >
[n
m+1 (ab2)n
m+2]
<la exi sha1_base64="Db51iQ8zL//n Js7 Xn1X3T/ps=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Q h79qFW e 5Z C 2BNY8caakiKao++ZHpx jlJNIYQZSuo6dKC8DoShmZFTopJIkgA QI66mEXAi WzSZ2QdhLGwVJ9Yk/ bAZcyiEPdIaD6s Zbyz+57mpCi+8jEZJqkiEdUR7YcosFV jWawuFQQ N QEsKD6Sg 3QQBWe yC u/Mlp0nzW FOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPx Px8h3NGdM/u+gPjPc yUeY+A==</la exi >
[n
m+1 (ab2)n
m+2]
<la exi sha1_base64="Db51iQ8zL//n Js7 Xn1X3T/ps=">AAACCXicbVDNSsNAGNzU 1 /oh69BI SkZakinosePFYw 5AGsOX7aZdupuE3Y1QQp9AX0ZPo 68+gK+jd aUF nNDszC99MkDAqlW1/G mFxaXll xqYW19Y3PL3N5pyjgVmDRwzGLRDkASRiPSUFQx0k4EAR4w0goGl2O/dUeEpHF0o4YJ8Tj0IhpSDEpL nnodn AO gZP3ZG 1G5BH5Q h79qFW e 5Z C 2BNY8caakiKao++ZHpx jlJNIYQZSuo6dKC8DoShmZFTopJIkgA QI66mEXAi WzSZ2QdhLGwVJ9Yk/ bAZcyiEPdIaD6s Zbyz+57mpCi+8jEZJqkiEdUR7YcosFV jWawuFQQ N QEsKD6Sg 3QQBWe yC u/Mlp0nzW FOaucXJ8Wa6XpEHm0h/ZRCTnoHNXQFaqjBsLoAT2hV/Rm3BuPx Px8h3NGdM/u+gPjPc yUeY+A==</la exi >
[n
m+1 (ab2)n
m+2]
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1
ab b
Figu e 3. Kinds o ows and main diagonal. S ep 2.
21

•Fo b= 1, ..., p,
2γn
m+1 −γn
m+2(ab−2) = [γn
m+1 −(ab−1)γn
m+2]+[γn
m+1 +γn
m+2].
•Fo b=p+ 1, ..., p +qand ib,
2γn
m+1 −( b−2)γn
m+2 = [γn
m+1 −( b−1)γn
m+2]+[γn
m+1 +γn
m+2].
•Fo b=p+ 1, ..., p +qand jb,
1
ab− b
[γn
m+1 −(ab− b−1)γn
m+2] = [−γn
m+2] + 1
ab− b
[γn
m+1 +γn
m+2].
•Fo b=p+q+ 1, ..., m and ib,
γn
m+1 = [−γn
m+2]+[γn
m+1 +γn
m+2].
•Fo b=p+q+ 1, ..., m and jb,
2γn
m+1 −γn
m+2(ab−3) = [γn
m+1 −(ab−2)γn
m+2]+[γn
m+1 +γn
m+2].
Following equali y (12), we ob ain he de e minan
D2m−p=γn
m+1 +γn
m+22m−p
Qp+q
k=p+1(ak− k)
1 +
p
X
k=1
γn
m+1 −(ak−1)γn
m+2
γn
m+1 +γn
m+2
+
p+q
X
k=p+1
γn
m+1 −( k−1)γn
m+2
γn
m+1 +γn
m+2
+
p+q
X
k=p+1
−γn
m+2(ak− k)
γn
m+1 +γn
m+2
+
m
X
k=p+q+1
−γn
m+2
γn
m+1 +γn
m+2
+
m
X
k=p+q+1
γn
m+1 −(ak−2)γn
m+2
γn
m+1 +γn
m+2 

=γn
m+1 +γn
m+22m−p−1
Qp+q
k=p+1(ak− k)(m+ 1)γn
m+1 + (m−n+ 1)γn
m+2.
I is easy o check ha D2m−p= 0, because
(m+ 1)γn
m+1 + (m−n+ 1)γn
m+2 = 0.
Bu he e is a condi ion on he payo s o P, e iciency, ha we ha e no used. We only
needed o e i y 2m−p−1 equali ies, ha is, we can conside he same de e minan
wi h one less equa ion and one less unknown. I he e exis s kwi h ak6= k hen we
dele e ow kand, by (12) (i can be mul iplied by ak− k),
Dk
2m−p−1=γn
m+2(ak− k)(γn
m+1 +γn
m+2)2m−p−2
Qp+q
b=p+1,b6=k(ab− b)6= 0,
whe e Dk
2m−p−1is he de e minan ob ained when we dele e ow and column kin
D2m−p. Now we suppose ak= k o all k= 1, ..., m. In his case
Dk
2m−p−1= (γn
m+1 −(ak−1)γn
m+2)(γn
m+1 +γn
m+2)2m−p−2.
22
I he e is k∈ {1, ..., m}wi h γn
m+1 −(ak−1)γn
m+2 6= 0, hen we dele e ow kand we
ge Dk
2m−p−16= 0. I γn
m+1 −(ak−1)γn
m+2 = 0 o all k= 1, ..., m, hen all he g oups
ha e he same size ak=n
m+1.Bu i all he g oups ha e he same size a hen am =n
and his implies a=n
m.
Funding
This esea ch has been suppo ed by he FQM237 g an o he Andalusian Go e nmen
Re e ences
[1] Albizu i M.J., Au ekoe xea J. and Za zuelo J.M. Con igu a ion alues: ex ensions o he
coali ional Owen alue. Games and Economic Beha io ; 2006 57: 1–17.
[2] Alonso-Meijide, J.M. and Ca e as, F. The p opo ional coali ional Shapley alue. Expe
Sys ems wi h Applica ions; 2011 38: 6967-6979.
[3] Alonso-Meijide, J.M., Casas-M´endez, B., Gonz´alez-Rueda, M. and Lo enzo-F ei e, S. Ax-
ioma ic o he Shapley alue o a game wi h a p io i unions. Top; 2014, 22(2): 749–770.
[4] B ink, R. an den and Die z, C. Union alues o games wi h coali ion s uc u e. Decision
Suppo Sys ems; 2014, 66: 1-8.
[5] Fe n´andez, J.R. , Gallego, I., Jim´enez-Losada A. and O d´o˜nez, M. Coope a ion among
agen s wi h a p oximi y ela ion. Eu opean Jou nal o Ope a ional Resea ch; 2016, 250:
555-565.
[6] Fe n´andez, J.R., Gallego, I., Jim´enez-Losada, A. and O d´o˜nez, M. A Banzha alue o
games wi h a p oximi y ela ion among he agen s. In e na ional Jou nal o App oxima e
Reasoning; 2017, 88: 192-208.
[7] G´omez-R´ua, M. and Vidal-Puga, J. The axioma ic app oach o h ee alues in games
wi h coali ion s uc u e. Eu opean Jou nal o Ope a ional Resea ch; 2010, 207: 795-806.
[8] Gon¸cal es-Dosan os, J.C., Alonso-Meijide J.M. New esul s on egali a ian alues o
games wi h a p io i unions. Op imiza ion; 2021, 67.
[9] Hue ne , F. and Wiese, H. The need o pe mission, he powe o en o ce, and duali y
in coope a i e games wi h a hie a chy. In e na ional Game Theo y Re iew; 2016, 18(4).
[10] He be , J.P. and Yao, J.T. Game- heo e ic ough se s. Fundamen a In o ma icae; 2011,
108(3–4):267–286.
[11] Kamijo, Y. A wo-s ep Shapley alue o coope a i e games wi h coali ion s uc u es.
In e na ional Game Theo y Re iew; 2009, 11(2):207-214.
[12] O d´o˜nez, M., Fe n´andez, J.R., Gallego, I. and Jim´enez-Losada, A. A alue o games on
colo ed communica ion s uc u es. Ope a ional Resea ch; 2021, 21: 2247-2260.
[13] Owen, G. Values o Games wi h a P io i Unions. In: Henn R., Moeschlin O. (eds) Ma h-
ema ical Economics and Game Theo y. Lec u e No es in Economics and Ma hema ical
Sys ems, ol 141. Sp inge , Be lin, Heidelbe g. 1977.
[14] Pawlak, Z. Rough se s. In e na ional Jou nal o Pa allel P og amming; 1982, 11(5):
341–356.
[15] Polkowski, L. and A aszkiewicz, B. A ough se app oach o es ima ing he game alue
and he Shapley alue om da a. Fundamen a In o ma icae; 2002, 82(4): 219–227.
[16] Saa ed a-Nie es, A. and Fies as-Janei o, G.M. Sampling me hods o es ima e he
Banzha –Owen alue. Annals o Ope a ions Resea ch; 2021, 301(1-2): 199-223.
[17] Shapley, L.S. A Value o n-pe son Games. In Con ibu ions o he Theo y o Games,
olume II, by H.W. Kuhn and A.W. Tucke , edi o s. Annals o Ma hema ical S udies .
28, P ince on Uni e si y P ess: 307–317, 1953.
[18] Radzik T., D iessen T. On a amily o alues o TU-games gene alizing he Shapley alue.
Ma hema ical Social Sciences; 2013, 65: 105–111.
23
Appendix A. Explica ion o Figu e 2
We explain he e he i e di e en kinds o ows showed in Figu e 2.
1) Conside an equa ion co esponding o b∈ {1, ..., p}in (15). Take DH∪{i}wi h
i /∈Ab. Obse e ha i he e a e Ksymme ic (in P) coali ions Scon aining ib
o i in he equa ion, hen he e a e also (ab−1)Ksymme ic S {ib}op ions
o i in he equa ion. So, he coe icien o numbe αDH∪{i}
His
Kγn
m+1 −γn
m+2(ab−1)(A1)
Le X1
b=
p
Y
k=1
k6=b
ak
p+q
Y
k=p+1
k
m
Y
k=p+q+1
(ak−1). We look o he coe icien o he chosen
numbe s in his equa ion, analyzing he i e kinds o numbe s plus he numbe
co esponding o he equa ion.
1.1 Numbe αDH∪{iH
b}
H(we canno apply (A1) he e). I appea s o all he coali-
ions equi alen by symme y o S=DH∪{iH
b}, namely
(ab−1)
p
Y
k=1
k6=b
ak
p+q
Y
k=p+1
k
m
Y
k=p+q+1
(ak−1)
imes, and wi h coali ions equi alen by symme y o S {ib}=DH∪{iH
b},
namely
ab−1
2p
Y
k=1
k6=b
ak
p+q
Y
k=p+1
k
m
Y
k=p+q+1
(ak−1)
imes. So, he coe icien o his numbe is
X1
b
ab−1
22γn
m+1 −(ab−2)γn
m+2.
1.2 Numbe αDH∪{iH
c}
H, wi h c∈ {1, ..., p} {b}. The quan i y Ko symme ic
coali ions (compa ible in P)Scon aining ib o DH∪{iH
c}is
K=ac
2p
Y
k=1
k6=b,c
ak
p+q
Y
k=p+1
k
m
Y
k=p+q+1
(ak−1).
Since (A1) we ge he coe icien o his numbe in he equa ion
X1
b
ac−1
2γn
m+1 −(ab−1)γn
m+2.
24
1.3 Numbe αDH∪{iH
c}
H,c∈ {p+ 1, ..., p +q}. We ob ain in his case
K= c
2p
Y
k=1
k6=b
ak
p+q
Y
k=p+1
k6=c
k
m
Y
k=p+q+1
(ak−1).
Since (A1) we ge he coe icien o his numbe
X1
b
c−1
2γn
m+1 −(ab−1)γn
m+2.
1.4 Numbe αDH∪{jH
c}
H,c∈ {p+ 1, ..., p +q}. We ge
K= c(ac− c)
p
Y
k=1
k6=b
ak
p+q
Y
k=p+1
k6=c
k
m
Y
k=p+q+1
(ak−1),
and hen, om (A1), he coe icien o he numbe is
X1
b(ac− c)γn
m+1 −(ab−1)γn
m+2.
1.5 Numbe αDH∪{iH
c}
H,c∈ {p+q+ 1, ..., m}. I holds
K=
p
Y
k=1
k6=b
ak
p+q
Y
k=p+1
k
m
Y
k=p+q+1
(ak−1).
Since (A1) he coe icien o he numbe is
X1
bγn
m+1 −(ab−1)γn
m+2.
1.6 Numbe αDH∪{jH
c}
H,c∈ {p+q+ 1, ..., m}. We ge
K=ac−1
2p
Y
k=1
k6=b
ak
p+q
Y
k=p+1
k
m
Y
k=p+q+1
k6=c
(ak−1).
Using (A1), he coe icien o he numbe is
X1
b
ac−2
2γn
m+1 −(ab−1)γn
m+2.
2) Conside an equa ion co esponding o b∈ {p+1, ..., p+q}in (15). Take DH∪{i}
wi h i /∈Tb. Obse e ha i he e a e Ksymme ic (in P) coali ions Scon aining
ib o i in he equa ion, hen he e a e also ( b−1)Ksymme ic S {ib}op ions
o i in he equa ion. So, he coe icien o numbe αDH∪{i}
His
Kγn
m+1 −γn
m+2( b−1)(A2)
25